@article{
 title = {Effective anomalous diffusion in a conical channel},
 type = {article},
 year = {2025},
 volume = {35},
 id = {97cbd421-d2fb-3fc4-9f69-3ae8bd6c274c},
 created = {2025-02-25T09:27:46.280Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2025-02-25T09:27:46.280Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {true},
 abstract = {Transport through structures such as pores and ion channels is ubiquitous in nature. It has been intensively studied in recent years. Especially in biological cells, the movement of molecules through channel systems plays an essential role in controlling almost every physiological function of living organisms. The subject of our study is the kinetics of spherical particles passing through a conical pore restricted by absorbing and reflecting boundaries from a wider to a narrower end and vice versa. We study the properties of diffusion as a function of particle size with respect to pore width. Particles of different diameters are subjected to a random force. In addition to the mean squared displacement, which indicates the (effective) subdiffusive or superdiffusive character of the motion (depending on whether the absorbing boundary is located at the narrow or wide end of the channel), we measured the mean and median of the first passage times. Additional in silico experiments allowed us to thoroughly discuss the interplay of entropic forces and boundary conditions influencing the obtained results.},
 bibtype = {article},
 author = {Cieśla, M. and Dybiec, B. and Krasowska, M. and Strzelewicz, A.},
 doi = {10.1063/5.0243989},
 journal = {Chaos},
 number = {2}
}

Transport through structures such as pores and ion channels is ubiquitous in nature. It has been intensively studied in recent years. Especially in biological cells, the movement of molecules through channel systems plays an essential role in controlling almost every physiological function of living organisms. The subject of our study is the kinetics of spherical particles passing through a conical pore restricted by absorbing and reflecting boundaries from a wider to a narrower end and vice versa. We study the properties of diffusion as a function of particle size with respect to pore width. Particles of different diameters are subjected to a random force. In addition to the mean squared displacement, which indicates the (effective) subdiffusive or superdiffusive character of the motion (depending on whether the absorbing boundary is located at the narrow or wide end of the channel), we measured the mean and median of the first passage times. Additional in silico experiments allowed us to thoroughly discuss the interplay of entropic forces and boundary conditions influencing the obtained results.

@article{
 title = {Multimodality in systems driven by Ornstein-Uhlenbeck noise},
 type = {article},
 year = {2024},
 volume = {34},
 id = {3d3b4455-a5e7-3e76-a909-0243db2657b4},
 created = {2025-02-25T09:27:46.279Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2025-02-25T09:27:46.279Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {true},
 abstract = {The presence of noise in nonlinear dynamical systems can significantly change their properties. Here, we study the properties of a noise perturbed motion in a single-well potential of | x | n ( n > 0 ) type. We explore under what conditions the action of the Ornstein-Uhlenbeck noise induces bimodality of stationary states in static, single-well, power-law potentials. In particular, we inspect the transition from unimodality ( n ⩽ 2 ) to bimodality ( n > 2 ). Results of numerical simulations are compared with estimates obtained from the unified colored-noise approximation. Furthermore, we explore the role of a harmonic addition to the general single-well power-law potentials showing its constructive or destructive role.},
 bibtype = {article},
 author = {Dybiec, B.},
 doi = {10.1063/5.0228666},
 journal = {Chaos},
 number = {11}
}

The presence of noise in nonlinear dynamical systems can significantly change their properties. Here, we study the properties of a noise perturbed motion in a single-well potential of | x | n ( n > 0 ) type. We explore under what conditions the action of the Ornstein-Uhlenbeck noise induces bimodality of stationary states in static, single-well, power-law potentials. In particular, we inspect the transition from unimodality ( n ⩽ 2 ) to bimodality ( n > 2 ). Results of numerical simulations are compared with estimates obtained from the unified colored-noise approximation. Furthermore, we explore the role of a harmonic addition to the general single-well power-law potentials showing its constructive or destructive role.

@article{
 title = {Role of long jumps in Lévy noise-induced multimodality},
 type = {article},
 year = {2024},
 volume = {34},
 id = {55f08e9e-535f-3810-8e3a-e88d53535df0},
 created = {2025-02-25T09:27:46.360Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2025-02-25T09:27:46.360Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {true},
 abstract = {Lévy noise is a paradigmatic noise used to describe out-of-equilibrium systems. Typically, properties of Lévy noise driven systems are very different from their Gaussian white noise driven counterparts. In particular, under action of Lévy noise, stationary states in single-well, super-harmonic, potentials are no longer unimodal. Typically, they are bimodal; however, for fine-tuned potentials, the number of modes can be further increased. The multimodality arises as a consequence of the competition between long displacements induced by the non-equilibrium stochastic driving and action of the deterministic force. Here, we explore robustness of bimodality in the quartic potential under action of the Lévy noise. We explore various scenarios of bounding long jumps and assess their ability to weaken and destroy multimodality. In general, we demonstrate that despite its robustness it is possible to destroy the bimodality, however it requires drastic reduction in the length of noise-induced jumps.},
 bibtype = {article},
 author = {Pogorzelec, P. and Dybiec, B.},
 doi = {10.1063/5.0206970},
 journal = {Chaos},
 number = {6}
}

Lévy noise is a paradigmatic noise used to describe out-of-equilibrium systems. Typically, properties of Lévy noise driven systems are very different from their Gaussian white noise driven counterparts. In particular, under action of Lévy noise, stationary states in single-well, super-harmonic, potentials are no longer unimodal. Typically, they are bimodal; however, for fine-tuned potentials, the number of modes can be further increased. The multimodality arises as a consequence of the competition between long displacements induced by the non-equilibrium stochastic driving and action of the deterministic force. Here, we explore robustness of bimodality in the quartic potential under action of the Lévy noise. We explore various scenarios of bounding long jumps and assess their ability to weaken and destroy multimodality. In general, we demonstrate that despite its robustness it is possible to destroy the bimodality, however it requires drastic reduction in the length of noise-induced jumps.

@article{
 title = {Numerical Modeling of Anisotropic Particle Diffusion through a Cylindrical Channel},
 type = {article},
 year = {2024},
 keywords = {artificial pore,channels and pores,first passage time,numerical modeling,overdamped diffusion,stochastic dynamics},
 volume = {29},
 id = {05976226-9809-3598-8d44-3d79c33e3af0},
 created = {2025-02-25T09:27:46.361Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2025-02-25T09:27:46.361Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {true},
 abstract = {The transport of molecules and particles through single pores is the basis of biological processes, including DNA and protein sequencing. As individual objects pass through a pore, they cause a transient change in the current that can be correlated with the object size, surface charge, and even chemical properties. The majority of experiments and modeling have been performed with spherical objects, while much less is known about the transport characteristics of aspherical particles, which would act as a model system, for example, for proteins and bacteria. The transport kinetics of aspherical objects is an especially important, yet understudied, problem in nanopore analytics. Here, using the Wiener process, we present a simplified model of the diffusion of rod-shaped particles through a cylindrical pore, and apply it to understand the translation and rotation of the particles as they pass through the pore. Specifically, we analyze the influence of the particles’ geometrical characteristics on the effective diffusion type, the first passage time distribution, and the particles’ orientation in the pore. Our model shows that thicker particles pass through the channel slower than thinner ones, while their lengths do not affect the passage time. We also demonstrate that both spherical and rod-shaped particles undergo normal diffusion, and the first passage time distribution follows an exponential asymptotics. The model provides guidance on how the shape of the particle can be modified to achieve an optimal passage time.},
 bibtype = {article},
 author = {Cieśla, M. and Dybiec, B. and Krasowska, M. and Siwy, Z.S. and Strzelewicz, A.},
 doi = {10.3390/molecules29163795},
 journal = {Molecules},
 number = {16}
}

The transport of molecules and particles through single pores is the basis of biological processes, including DNA and protein sequencing. As individual objects pass through a pore, they cause a transient change in the current that can be correlated with the object size, surface charge, and even chemical properties. The majority of experiments and modeling have been performed with spherical objects, while much less is known about the transport characteristics of aspherical particles, which would act as a model system, for example, for proteins and bacteria. The transport kinetics of aspherical objects is an especially important, yet understudied, problem in nanopore analytics. Here, using the Wiener process, we present a simplified model of the diffusion of rod-shaped particles through a cylindrical pore, and apply it to understand the translation and rotation of the particles as they pass through the pore. Specifically, we analyze the influence of the particles’ geometrical characteristics on the effective diffusion type, the first passage time distribution, and the particles’ orientation in the pore. Our model shows that thicker particles pass through the channel slower than thinner ones, while their lengths do not affect the passage time. We also demonstrate that both spherical and rod-shaped particles undergo normal diffusion, and the first passage time distribution follows an exponential asymptotics. The model provides guidance on how the shape of the particle can be modified to achieve an optimal passage time.

@article{
 title = {Lévy flights and Lévy walks under stochastic resetting},
 type = {article},
 year = {2024},
 volume = {109},
 id = {ae50d54f-0ddd-3cde-91a8-aa2a249239e3},
 created = {2025-02-25T09:27:46.466Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2025-02-25T09:27:46.466Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {true},
 abstract = {Stochastic resetting is a protocol of starting anew, which can be used to facilitate the escape kinetics. We demonstrate that restarting can accelerate the escape kinetics from a finite interval restricted by two absorbing boundaries also in the presence of heavy-tailed, Lévy-type, α-stable noise. However, the width of the domain where resetting is beneficial depends on the value of the stability index α determining the power-law decay of the jump length distribution. For heavier (smaller α) distributions, the domain becomes narrower in comparison to lighter tails. Additionally, we explore connections between Lévy flights (LFs) and Lévy walks (LWs) in the presence of stochastic resetting. First of all, we show that for Lévy walks, the stochastic resetting can also be beneficial in the domain where the coefficient of variation is smaller than 1. Moreover, we demonstrate that in the domain where LWs are characterized by a finite mean jump duration (length), with the increasing width of the interval, the LWs start to share similarities with LFs under stochastic resetting. },
 bibtype = {article},
 author = {Zbik, B. and Dybiec, B.},
 doi = {10.1103/PhysRevE.109.044147},
 journal = {Physical Review E},
 number = {4}
}

Stochastic resetting is a protocol of starting anew, which can be used to facilitate the escape kinetics. We demonstrate that restarting can accelerate the escape kinetics from a finite interval restricted by two absorbing boundaries also in the presence of heavy-tailed, Lévy-type, α-stable noise. However, the width of the domain where resetting is beneficial depends on the value of the stability index α determining the power-law decay of the jump length distribution. For heavier (smaller α) distributions, the domain becomes narrower in comparison to lighter tails. Additionally, we explore connections between Lévy flights (LFs) and Lévy walks (LWs) in the presence of stochastic resetting. First of all, we show that for Lévy walks, the stochastic resetting can also be beneficial in the domain where the coefficient of variation is smaller than 1. Moreover, we demonstrate that in the domain where LWs are characterized by a finite mean jump duration (length), with the increasing width of the interval, the LWs start to share similarities with LFs under stochastic resetting.

@article{
 title = {Weighted Axelrod model: Different but similar},
 type = {article},
 year = {2023},
 keywords = {Agent based modeling,Axelrod model,Extended conservativeness,Social dynamics},
 volume = {630},
 id = {61a30a93-dde2-325f-9638-ea6ebd9d7aff},
 created = {2025-02-25T09:27:46.486Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2025-02-25T09:27:46.486Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {true},
 abstract = {The Axelrod model is a cellular automaton which can be used to describe the emergence and development of cultural domains, where culture is represented by a fixed number of cultural features taking a discrete set of possible values (traits). The Axelrod model is based on two sociological phenomena: homophily (a tendency for individuals to form bonds with people similar to themselves) and social influence (the way how individuals change their behavior due to social pressure). However, the Axelrod model does not take into account the fact that cultural attributes may have different significance for a given individual. This is a limitation in the context of how the model reflects mechanisms driving the evolution of real societies. The study aims to modify the Axelrod model by giving individual features different weights that have a decisive impact on the possibility of aligning cultural traits between (interacting) individuals. The comparison of the results obtained for the classic Axelrod model and its modified version shows that introduced weights have a significant impact on the course of the system development, in particular, increasing the final fragmentation of the system and increasing the time needed to reach the final state.},
 bibtype = {article},
 author = {Kalinowska, Z. and Dybiec, B.},
 doi = {10.1016/j.physa.2023.129281},
 journal = {Physica A: Statistical Mechanics and its Applications}
}

The Axelrod model is a cellular automaton which can be used to describe the emergence and development of cultural domains, where culture is represented by a fixed number of cultural features taking a discrete set of possible values (traits). The Axelrod model is based on two sociological phenomena: homophily (a tendency for individuals to form bonds with people similar to themselves) and social influence (the way how individuals change their behavior due to social pressure). However, the Axelrod model does not take into account the fact that cultural attributes may have different significance for a given individual. This is a limitation in the context of how the model reflects mechanisms driving the evolution of real societies. The study aims to modify the Axelrod model by giving individual features different weights that have a decisive impact on the possibility of aligning cultural traits between (interacting) individuals. The comparison of the results obtained for the classic Axelrod model and its modified version shows that introduced weights have a significant impact on the course of the system development, in particular, increasing the final fragmentation of the system and increasing the time needed to reach the final state.

@article{
 title = {Optimization of escape kinetics by reflecting and resetting},
 type = {article},
 year = {2023},
 volume = {33},
 id = {3ef5fbfd-5348-36a6-8eb1-9903212839a4},
 created = {2025-02-25T09:27:46.533Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2025-02-25T09:27:46.533Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {true},
 abstract = {Stochastic restarting is a strategy of starting anew. Incorporation of the resetting to the random walks can result in a decrease in the mean first passage time due to the ability to limit unfavorably meandering, sub-optimal trajectories. In this paper, we examine how stochastic resetting influences escape dynamics from the ( − ∞ , 1 ) interval in the presence of the single-well power-law | x | κ potentials with κ > 0 . Examination of the mean first passage time is complemented by the analysis of the coefficient of variation, which provides a robust and reliable indicator assessing the efficiency of stochastic resetting. The restrictive nature of resetting is compared to placing a reflective boundary in the system at hand. In particular, for each potential, the position of the reflecting barrier giving the same mean first passage time as the optimal resetting rate is determined. Finally, in addition to reflecting, we compare the effectiveness of other resetting strategies with respect to optimization of the mean first passage time.},
 bibtype = {article},
 author = {Capała, K. and Dybiec, B.},
 doi = {10.1063/5.0159475},
 journal = {Chaos},
 number = {10}
}

Stochastic restarting is a strategy of starting anew. Incorporation of the resetting to the random walks can result in a decrease in the mean first passage time due to the ability to limit unfavorably meandering, sub-optimal trajectories. In this paper, we examine how stochastic resetting influences escape dynamics from the ( − ∞ , 1 ) interval in the presence of the single-well power-law | x | κ potentials with κ > 0 . Examination of the mean first passage time is complemented by the analysis of the coefficient of variation, which provides a robust and reliable indicator assessing the efficiency of stochastic resetting. The restrictive nature of resetting is compared to placing a reflective boundary in the system at hand. In particular, for each potential, the position of the reflecting barrier giving the same mean first passage time as the optimal resetting rate is determined. Finally, in addition to reflecting, we compare the effectiveness of other resetting strategies with respect to optimization of the mean first passage time.

@article{
 title = {Resetting induced multimodality},
 type = {article},
 year = {2023},
 volume = {33},
 id = {62f1b4c9-5e5e-3079-856a-7d57395fe0e9},
 created = {2025-02-25T09:27:46.550Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2025-02-25T09:27:46.550Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {true},
 abstract = {Properties of stochastic systems are defined by the noise type and deterministic forces acting on the system. In out-of-equilibrium setups, e.g., for motions under action of Lévy noises, the existence of the stationary state is not only determined by the potential but also by the noise. Potential wells need to be steeper than parabolic in order to assure the existence of stationary states. The existence of stationary states, in sub-harmonic potential wells, can be restored by stochastic resetting, which is the protocol of starting over at random times. Herein, we demonstrate that the combined action of Lévy noise and Poissonian stochastic resetting can result in the phase transition between non-equilibrium stationary states of various multimodality in the overdamped system in super-harmonic potentials. Fine-tuned resetting rates can increase the modality of stationary states, while for high resetting rates, the multimodality is destroyed as the stochastic resetting limits the spread of particles.},
 bibtype = {article},
 author = {Pogorzelec, P. and Dybiec, B.},
 doi = {10.1063/5.0143821},
 journal = {Chaos},
 number = {6}
}

Properties of stochastic systems are defined by the noise type and deterministic forces acting on the system. In out-of-equilibrium setups, e.g., for motions under action of Lévy noises, the existence of the stationary state is not only determined by the potential but also by the noise. Potential wells need to be steeper than parabolic in order to assure the existence of stationary states. The existence of stationary states, in sub-harmonic potential wells, can be restored by stochastic resetting, which is the protocol of starting over at random times. Herein, we demonstrate that the combined action of Lévy noise and Poissonian stochastic resetting can result in the phase transition between non-equilibrium stationary states of various multimodality in the overdamped system in super-harmonic potentials. Fine-tuned resetting rates can increase the modality of stationary states, while for high resetting rates, the multimodality is destroyed as the stochastic resetting limits the spread of particles.

@article{
 title = {Stochastic kinetics under combined action of two noise sources},
 type = {article},
 year = {2023},
 volume = {107},
 id = {60ef200a-76eb-3a35-b2b9-42722ceb4d6f},
 created = {2025-02-25T09:27:46.594Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2025-02-25T09:27:46.594Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {true},
 abstract = {We are exploring two archetypal noise-induced escape scenarios: Escape from a finite interval and from the positive half-line under the action of the mixture of Lévy and Gaussian white noises in the overdamped regime, for the random acceleration process and higher-order processes. In the case of escape from finite intervals, the mixture of noises can result in the change of value of the mean first passage time in comparison to the action of each noise separately. At the same time, for the random acceleration process on the (positive) half-line, over the wide range of parameters, the exponent characterizing the power-law decay of the survival probability is equal to the one characterizing the decay of the survival probability under action of the (pure) Lévy noise. There is a transient region, the width of which increases with stability index α, when the exponent decreases from the one for Lévy noise to the one corresponding to the Gaussian white noise driving. },
 bibtype = {article},
 author = {Pogorzelec, P. and Dybiec, B.},
 doi = {10.1103/PhysRevE.107.044124},
 journal = {Physical Review E},
 number = {4}
}

We are exploring two archetypal noise-induced escape scenarios: Escape from a finite interval and from the positive half-line under the action of the mixture of Lévy and Gaussian white noises in the overdamped regime, for the random acceleration process and higher-order processes. In the case of escape from finite intervals, the mixture of noises can result in the change of value of the mean first passage time in comparison to the action of each noise separately. At the same time, for the random acceleration process on the (positive) half-line, over the wide range of parameters, the exponent characterizing the power-law decay of the survival probability is equal to the one characterizing the decay of the survival probability under action of the (pure) Lévy noise. There is a transient region, the width of which increases with stability index α, when the exponent decreases from the one for Lévy noise to the one corresponding to the Gaussian white noise driving.

@article{
 title = {Escape from the potential well: accelerating by shaping and noise tuning},
 type = {article},
 year = {2023},
 keywords = {Lévy noise,mean first passage time,noise induced escape,stochastic dynamics},
 volume = {98},
 id = {8e60fcdf-1781-382f-8634-50206b9a072c},
 created = {2025-02-25T09:27:46.598Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2025-02-25T09:27:46.598Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {true},
 abstract = {Noise driven escape from the potential well is the basic component of various noise induced effects. The efficiency of the escape process or time scales matching is responsible for occurrence of the stochastic resonance and (stochastic) resonant activation. Here, we are extending the discussion on how the structure of the potential can be used to optimize the mean first passage time. It is demonstrated that corrugation of the potential can be beneficial under action of the weak Gaussian white noise. Furthermore, we show that the noise tuning can be more effective than shaping the potential. Therefore, action of the tuned additive α-stable noise can accelerate the escape kinetics more than corrugation of the potential. Finally, we demonstrate that mean first passage time from a potential well can be a non-monotonous function of the stability index α.},
 bibtype = {article},
 author = {Dybiec, B. and Zawiślak, M.},
 doi = {10.1088/1402-4896/acb46b},
 journal = {Physica Scripta},
 number = {2}
}

Noise driven escape from the potential well is the basic component of various noise induced effects. The efficiency of the escape process or time scales matching is responsible for occurrence of the stochastic resonance and (stochastic) resonant activation. Here, we are extending the discussion on how the structure of the potential can be used to optimize the mean first passage time. It is demonstrated that corrugation of the potential can be beneficial under action of the weak Gaussian white noise. Furthermore, we show that the noise tuning can be more effective than shaping the potential. Therefore, action of the tuned additive α-stable noise can accelerate the escape kinetics more than corrugation of the potential. Finally, we demonstrate that mean first passage time from a potential well can be a non-monotonous function of the stability index α.

@article{
 title = {Interplay of noise induced stability and stochastic resetting},
 type = {article},
 year = {2022},
 volume = {32},
 id = {daa2c66d-1130-3162-a4f6-fbc17cda917a},
 created = {2025-02-25T09:27:46.632Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2025-02-25T09:27:46.632Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {true},
 abstract = {Stochastic resetting and noise-enhanced stability are two phenomena that can affect the lifetime and relaxation of nonequilibrium states. They can be considered measures of controlling the efficiency of the completion process when a stochastic system has to reach the desired state. Here, we study the interaction of random (Poissonian) resetting and stochastic dynamics in unstable potentials. Unlike noise-induced stability that increases the relaxation time, the stochastic resetting may eliminate winding trajectories contributing to the lifetime and accelerate the escape kinetics from unstable states. In this paper, we present a framework to analyze compromises between the two contrasting phenomena in noise-driven kinetics subject to random restarts. },
 bibtype = {article},
 author = {Capała, K. and Dybiec, B. and Gudowska-Nowak, E.},
 doi = {10.1063/5.0092887},
 journal = {Chaos},
 number = {6}
}

Stochastic resetting and noise-enhanced stability are two phenomena that can affect the lifetime and relaxation of nonequilibrium states. They can be considered measures of controlling the efficiency of the completion process when a stochastic system has to reach the desired state. Here, we study the interaction of random (Poissonian) resetting and stochastic dynamics in unstable potentials. Unlike noise-induced stability that increases the relaxation time, the stochastic resetting may eliminate winding trajectories contributing to the lifetime and accelerate the escape kinetics from unstable states. In this paper, we present a framework to analyze compromises between the two contrasting phenomena in noise-driven kinetics subject to random restarts.

@article{
 title = {Drifted escape from the finite interval},
 type = {article},
 year = {2022},
 volume = {105},
 id = {6ab06e63-8b2c-3d96-91cf-13fe99e122ef},
 created = {2025-02-25T09:27:46.635Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2025-02-25T09:27:46.635Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {true},
 abstract = {Properties of the noise-driven escape kinetics are mainly determined by the stochastic component of the system dynamics. Nevertheless, the escape dynamics is also sensitive to deterministic forces. Here, we are exploring properties of the overdamped drifted escape from finite intervals under the action of symmetric α-stable noises. We show that the properly rescaled mean first passage time follows the universal pattern as a function of the generalized Pécklet number, which can be used to efficiently discriminate between domains where drift or random force dominate. Stochastic driving of the α-stable type is capable of diminishing the significance of the drift in the regime when the drift prevails.},
 bibtype = {article},
 author = {Capała, K. and Dybiec, B.},
 doi = {10.1103/PhysRevE.105.064125},
 journal = {Physical Review E},
 number = {6}
}

Properties of the noise-driven escape kinetics are mainly determined by the stochastic component of the system dynamics. Nevertheless, the escape dynamics is also sensitive to deterministic forces. Here, we are exploring properties of the overdamped drifted escape from finite intervals under the action of symmetric α-stable noises. We show that the properly rescaled mean first passage time follows the universal pattern as a function of the generalized Pécklet number, which can be used to efficiently discriminate between domains where drift or random force dominate. Stochastic driving of the α-stable type is capable of diminishing the significance of the drift in the regime when the drift prevails.

@article{
 title = {Diffusion in crowded environments: Trapped by the drift},
 type = {article},
 year = {2021},
 volume = {104},
 id = {1f7c8a82-7b4b-312c-920a-02acf732fc40},
 created = {2025-02-25T09:27:46.673Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2025-02-25T09:27:46.673Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {true},
 abstract = {The diffusion type is determined not only by microscopic dynamics but also by the environment properties. For example, the environment's fractal structure is responsible for the emergence of subdiffusive scaling of the mean square displacement in Markovian systems because the presence of nontrivially placed obstacles puts constraints on possible displacements. We investigate how the additional action of drift changes properties of the diffusion in the crowded environment. It is shown that the action of a constant drift increases chances of trapping, which suppresses the persistent ballistic motion. Such a diffusion becomes anisotropic because the drift introduces a preferred direction of motion which is further altered by interactions with obstacles. Moreover, individual trajectories display a high level of variability, which is responsible for the macroscopic properties of the diffusing front. Overall, the interplay among drift, diffusion, and a crowded environment, as measured by the time-averaged mean square displacement, is responsible for the emergence of superdiffusive and subdiffusive patterns in the very same system. Importantly, in contrast to free motion, the constant drift can enhance signatures of subdiffusive motion as it increases trapping chances.},
 bibtype = {article},
 author = {Kubala, P. and Cieśla, M. and Dybiec, B.},
 doi = {10.1103/PhysRevE.104.044127},
 journal = {Physical Review E},
 number = {4}
}

The diffusion type is determined not only by microscopic dynamics but also by the environment properties. For example, the environment's fractal structure is responsible for the emergence of subdiffusive scaling of the mean square displacement in Markovian systems because the presence of nontrivially placed obstacles puts constraints on possible displacements. We investigate how the additional action of drift changes properties of the diffusion in the crowded environment. It is shown that the action of a constant drift increases chances of trapping, which suppresses the persistent ballistic motion. Such a diffusion becomes anisotropic because the drift introduces a preferred direction of motion which is further altered by interactions with obstacles. Moreover, individual trajectories display a high level of variability, which is responsible for the macroscopic properties of the diffusing front. Overall, the interplay among drift, diffusion, and a crowded environment, as measured by the time-averaged mean square displacement, is responsible for the emergence of superdiffusive and subdiffusive patterns in the very same system. Importantly, in contrast to free motion, the constant drift can enhance signatures of subdiffusive motion as it increases trapping chances.

@article{
 title = {Random acceleration process on finite intervals under stochastic restarting},
 type = {article},
 year = {2021},
 keywords = {Diffusion,First passage,Stochastic particle dynamics,Stochastic processes},
 volume = {2021},
 id = {0c47b0fc-cbf5-344e-816d-7b9a3f979db6},
 created = {2025-02-25T09:27:46.675Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2025-02-25T09:27:46.675Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {true},
 abstract = {The escape of the randomly accelerated undamped particle from the finite interval under action of stochastic resetting is studied. The motion of such a particle is described by the full Langevin equation and the particle is characterized by the position and velocity. We compare three resetting protocols, which restarts velocity or position (partial resetting) and the whole motion (position and velocity—full resetting). Using the mean first passage time we assess efficiency of restarting protocols in facilitating or suppressing the escape kinetics. There are fundamental differences between partial resetting scenarios which restart velocity or position, as in the limit of very frequent resets only the position resetting (regardless of initial velocity) can trap the particle in the finite domain of motion. The velocity resetting or the simultaneous position and velocity restarting provide a possibility of facilitating the undamped escape kinetics.},
 bibtype = {article},
 author = {Capała, K. and Dybiec, B.},
 doi = {10.1088/1742-5468/ac1664},
 journal = {Journal of Statistical Mechanics: Theory and Experiment},
 number = {8}
}

The escape of the randomly accelerated undamped particle from the finite interval under action of stochastic resetting is studied. The motion of such a particle is described by the full Langevin equation and the particle is characterized by the position and velocity. We compare three resetting protocols, which restarts velocity or position (partial resetting) and the whole motion (position and velocity—full resetting). Using the mean first passage time we assess efficiency of restarting protocols in facilitating or suppressing the escape kinetics. There are fundamental differences between partial resetting scenarios which restart velocity or position, as in the limit of very frequent resets only the position resetting (regardless of initial velocity) can trap the particle in the finite domain of motion. The velocity resetting or the simultaneous position and velocity restarting provide a possibility of facilitating the undamped escape kinetics.

@article{
 title = {Inertial Lévy flights in bounded domains},
 type = {article},
 year = {2021},
 volume = {31},
 id = {7d69cdbe-5572-3a95-85be-ac3006dfbfa1},
 created = {2025-02-25T09:27:46.709Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2025-02-25T09:27:46.709Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {true},
 abstract = {The escape from a given domain is one of the fundamental problems in statistical physics and the theory of stochastic processes. Here, we explore properties of the escape of an inertial particle driven by Lévy noise from a bounded domain, restricted by two absorbing boundaries. The presence of two absorbing boundaries assures that the escape process can be characterized by the finite mean first passage time. The detailed analysis of escape kinetics shows that properties of the mean first passage time for the integrated Ornstein-Uhlenbeck process driven by Lévy noise are closely related to properties of the integrated Lévy motions, which, in turn, are close to properties of the integrated Wiener process. The extensive studies of the mean first passage time were complemented by examination of the escape velocity and energy along with their sensitivity to initial conditions.},
 bibtype = {article},
 author = {Capała, K. and Dybiec, B.},
 doi = {10.1063/5.0054634},
 journal = {Chaos},
 number = {8}
}

The escape from a given domain is one of the fundamental problems in statistical physics and the theory of stochastic processes. Here, we explore properties of the escape of an inertial particle driven by Lévy noise from a bounded domain, restricted by two absorbing boundaries. The presence of two absorbing boundaries assures that the escape process can be characterized by the finite mean first passage time. The detailed analysis of escape kinetics shows that properties of the mean first passage time for the integrated Ornstein-Uhlenbeck process driven by Lévy noise are closely related to properties of the integrated Lévy motions, which, in turn, are close to properties of the integrated Wiener process. The extensive studies of the mean first passage time were complemented by examination of the escape velocity and energy along with their sensitivity to initial conditions.

@article{
 title = {Physics of free climbing},
 type = {article},
 year = {2021},
 volume = {103},
 id = {019751c4-3be3-3b0f-9846-e2a3076352c6},
 created = {2025-02-25T09:27:46.714Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2025-02-25T09:27:46.714Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {true},
 abstract = {The theory of stochastic processes provides theoretical tools which can be efficiently used to explore the properties of noise-induced escape kinetics. Since noise-facilitated escape over the potential barrier resembles free climbing, one can use the first-passage time theory in an analysis of rock climbing. We perform the analysis of the mean first-passage time in order to answer the question regarding the optimal, i.e., resulting in the fastest climbing, rope length. It is demonstrated that there is a discrete set of favorable rope lengths assuring the shortest climbing times, as they correspond to local minima of mean first-passage time. Within the set of favorable rope lengths there is the optimal rope giving rise to the shortest climbing time. In particular, more experienced climbers can decrease their climbing time by using longer ropes.},
 bibtype = {article},
 author = {Dybiec, B. and Capała, K. and Barbasz, J.},
 doi = {10.1103/PhysRevE.103.062135},
 journal = {Physical Review E},
 number = {6}
}

The theory of stochastic processes provides theoretical tools which can be efficiently used to explore the properties of noise-induced escape kinetics. Since noise-facilitated escape over the potential barrier resembles free climbing, one can use the first-passage time theory in an analysis of rock climbing. We perform the analysis of the mean first-passage time in order to answer the question regarding the optimal, i.e., resulting in the fastest climbing, rope length. It is demonstrated that there is a discrete set of favorable rope lengths assuring the shortest climbing times, as they correspond to local minima of mean first-passage time. Within the set of favorable rope lengths there is the optimal rope giving rise to the shortest climbing time. In particular, more experienced climbers can decrease their climbing time by using longer ropes.

@article{
 title = {Dichotomous flow with thermal diffusion and stochastic resetting},
 type = {article},
 year = {2021},
 volume = {31},
 id = {49dd8f98-11b9-310b-9b72-a6c09dee2a5a},
 created = {2025-02-25T09:27:46.750Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2025-02-25T09:27:46.750Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {true},
 abstract = {We consider properties of one-dimensional diffusive dichotomous flow and discuss effects of stochastic resonant activation (SRA) in the presence of a statistically independent random resetting mechanism. Resonant activation and stochastic resetting are two similar effects, as both of them can optimize the noise-induced escape. Our studies show completely different origins of optimization in adapted setups. Efficiency of stochastic resetting relies on elimination of suboptimal trajectories, while SRA is associated with matching of time scales in the dynamic environment. Consequently, both effects can be easily tracked by studying their asymptotic properties. Finally, we show that stochastic resetting cannot be easily used to further optimize the SRA in symmetric setups.},
 bibtype = {article},
 author = {Capała, K. and Dybiec, B. and Gudowska-Nowak, E.},
 doi = {10.1063/5.0052789},
 journal = {Chaos},
 number = {6}
}

We consider properties of one-dimensional diffusive dichotomous flow and discuss effects of stochastic resonant activation (SRA) in the presence of a statistically independent random resetting mechanism. Resonant activation and stochastic resetting are two similar effects, as both of them can optimize the noise-induced escape. Our studies show completely different origins of optimization in adapted setups. Efficiency of stochastic resetting relies on elimination of suboptimal trajectories, while SRA is associated with matching of time scales in the dynamic environment. Consequently, both effects can be easily tracked by studying their asymptotic properties. Finally, we show that stochastic resetting cannot be easily used to further optimize the SRA in symmetric setups.

@article{
 title = {Deterministic force-free resonant activation},
 type = {article},
 year = {2021},
 keywords = {stochastic particle dynamics,stochastic processes},
 volume = {2021},
 id = {ec2343b0-48b9-3d67-aa52-8904ce77e3d9},
 created = {2025-02-25T09:27:46.755Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2025-02-25T09:27:46.755Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {true},
 abstract = {The combined action of noise and deterministic force in dynamical systems can induce resonant effects. Here, we demonstrate a minimal, deterministic force-free setup allowing for the occurrence of resonant, noise-induced effects. We show that in the archetypal problem of escape from finite intervals driven by α-stale noise with a periodically modulated stability index, depending on the initial direction of the modulation, resonant-activation-like or noise-enhanced-stability-like phenomena can be observed. Consequently, in comparison to traditional Lévy flights, Lévy flights with a time-dependent jump length exponent are capable of facilitating or slowing down the escape from finite intervals in an analogous way, such as the modulation of the potential in the resonant activation setup. },
 bibtype = {article},
 author = {Capała, K. and Dybiec, B.},
 doi = {10.1088/1742-5468/abd939},
 journal = {Journal of Statistical Mechanics: Theory and Experiment},
 number = {2}
}

The combined action of noise and deterministic force in dynamical systems can induce resonant effects. Here, we demonstrate a minimal, deterministic force-free setup allowing for the occurrence of resonant, noise-induced effects. We show that in the archetypal problem of escape from finite intervals driven by α-stale noise with a periodically modulated stability index, depending on the initial direction of the modulation, resonant-activation-like or noise-enhanced-stability-like phenomena can be observed. Consequently, in comparison to traditional Lévy flights, Lévy flights with a time-dependent jump length exponent are capable of facilitating or slowing down the escape from finite intervals in an analogous way, such as the modulation of the potential in the resonant activation setup.

@article{
 title = {Bounding energy growth in frictionless stochastic oscillators},
 type = {article},
 year = {2020},
 volume = {102},
 id = {dda2f4d7-21f9-346d-8db0-8e484f6708d6},
 created = {2020-10-30T10:12:15.025Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.025Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2020 American Physical Society. This paper presents analytical and numerical results on the energetics of nonharmonic, undamped, single-well, stochastic oscillators driven by additive Gaussian white noises. The absence of damping and the action of noise are responsible for the lack of stationary states in such systems. We explore the properties of average kinetic, potential, and total energies along with the generalized equipartition relations. It is demonstrated that in frictionless dynamics, nonequilibrium stationary states can be produced by stochastic resetting. For an appropriate resetting protocol, the average energies become bounded. If the resetting protocol is not characterized by a finite variance of renewal intervals, stochastic resetting can only slow down the growth of the average energies but it does not bound them. Under special conditions regarding the frequency of resets, the ratios of the average energies follow the generalized equipartition relations.},
 bibtype = {article},
 author = {Mandrysz, M. and Dybiec, B.},
 doi = {10.1103/PhysRevE.102.022105},
 journal = {Physical Review E},
 number = {2}
}

© 2020 American Physical Society. This paper presents analytical and numerical results on the energetics of nonharmonic, undamped, single-well, stochastic oscillators driven by additive Gaussian white noises. The absence of damping and the action of noise are responsible for the lack of stationary states in such systems. We explore the properties of average kinetic, potential, and total energies along with the generalized equipartition relations. It is demonstrated that in frictionless dynamics, nonequilibrium stationary states can be produced by stochastic resetting. For an appropriate resetting protocol, the average energies become bounded. If the resetting protocol is not characterized by a finite variance of renewal intervals, stochastic resetting can only slow down the growth of the average energies but it does not bound them. Under special conditions regarding the frequency of resets, the ratios of the average energies follow the generalized equipartition relations.

@article{
 title = {Energy partition for anharmonic, undamped, classical oscillators},
 type = {article},
 year = {2020},
 keywords = {Equipartition relations,anharmonic oscillators,classical oscillators,virial theorem},
 volume = {53},
 id = {af3123bf-e9fe-3eb4-a691-094a00c609c7},
 created = {2020-10-30T10:12:15.071Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.071Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2020 IOP Publishing Ltd. Using stochastic methods, general formulas for average kinetic and potential energies for anharmonic, undamped (frictionless), classical oscillators are derived. It is demonstrated that for potentials of |x|ν, (ν > 0) type energies are equipartitioned for the harmonic potential only. For potential wells weaker than parabolic potential energy dominates, while for potentials stronger than parabolic kinetic energy prevails. Due to energy conservation, the variances of kinetic and potential energies are equal. In the limiting case of the infinite rectangular potential well (ν → ∞) the whole energy is stored in the form of the kinetic energy and variances of energy distributions vanish.},
 bibtype = {article},
 author = {Mandrysz, M. and Dybiec, B.},
 doi = {10.1088/1751-8121/ab8135},
 journal = {Journal of Physics A: Mathematical and Theoretical},
 number = {19}
}

© 2020 IOP Publishing Ltd. Using stochastic methods, general formulas for average kinetic and potential energies for anharmonic, undamped (frictionless), classical oscillators are derived. It is demonstrated that for potentials of |x|ν, (ν > 0) type energies are equipartitioned for the harmonic potential only. For potential wells weaker than parabolic potential energy dominates, while for potentials stronger than parabolic kinetic energy prevails. Due to energy conservation, the variances of kinetic and potential energies are equal. In the limiting case of the infinite rectangular potential well (ν → ∞) the whole energy is stored in the form of the kinetic energy and variances of energy distributions vanish.

@article{
 title = {Nonlinear friction in underdamped anharmonic stochastic oscillators},
 type = {article},
 year = {2020},
 volume = {30},
 id = {9407fd47-4938-38c4-b50a-9618f239fa84},
 created = {2020-10-30T10:12:16.139Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.139Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {Non-equilibrium stationary states of overdamped anharmonic stochastic oscillators driven by Lévy noise are typically multimodal. The very same situation is recorded for an underdamped Lévy noise-driven motion in single-well potentials with linear friction. Within the current article, we relax the assumption that the friction experienced by a particle is linear. Using computer simulations, we study underdamped motions in single-well potentials in the regime of nonlinear friction. We demonstrate that it is relatively easy to observe multimodality in the velocity distribution as it is determined by the friction itself and it is the same as the multimodality in the overdamped case with the analogous deterministic force. Contrary to the velocity marginal density, it is more difficult to induce multimodality in the position. Nevertheless, for a fine-tuned nonlinear friction, the spatial multimodality can be recorded.},
 bibtype = {article},
 author = {Capała, K. and Dybiec, B. and Gudowska-Nowak, E.},
 doi = {10.1063/5.0007581},
 journal = {Chaos (Woodbury, N.Y.)},
 number = {7}
}

Non-equilibrium stationary states of overdamped anharmonic stochastic oscillators driven by Lévy noise are typically multimodal. The very same situation is recorded for an underdamped Lévy noise-driven motion in single-well potentials with linear friction. Within the current article, we relax the assumption that the friction experienced by a particle is linear. Using computer simulations, we study underdamped motions in single-well potentials in the regime of nonlinear friction. We demonstrate that it is relatively easy to observe multimodality in the velocity distribution as it is determined by the friction itself and it is the same as the multimodality in the overdamped case with the analogous deterministic force. Contrary to the velocity marginal density, it is more difficult to induce multimodality in the position. Nevertheless, for a fine-tuned nonlinear friction, the spatial multimodality can be recorded.

@article{
 title = {Peculiarities of escape kinetics in the presence of athermal noises},
 type = {article},
 year = {2020},
 volume = {30},
 id = {cac19cb8-cc74-3cfa-aba6-0850b47e2f56},
 created = {2020-10-30T10:12:16.189Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.189Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2020 Author(s). Stochastic evolution of various dynamic systems and reaction networks is commonly described in terms of noise assisted escape of an overdamped particle from a potential well, as devised by the paradigmatic Langevin equation in which additive Gaussian stochastic force reproduces effects of thermal fluctuations from the reservoir. When implemented for systems close to equilibrium, the approach correctly explains the emergence of the Boltzmann distribution for the ensemble of trajectories generated by the Langevin equation and relates the intensity of the noise strength to the mobility. This scenario can be further generalized to include effects of non-Gaussian, burstlike forcing modeled by Lévy noise. In this case, however, the pulsatile additive noise cannot be treated as the internal (thermal) since the relation between the strength of the friction and variance of the noise is violated. Heavy tails of Lévy noise distributions not only facilitate escape kinetics, but also, more importantly, change the escape protocol by altering the final stationary state to a non-Boltzmann, nonequilibrium form. As a result, contrary to the kinetics induced by a Gaussian white noise, escape rates in environments with Lévy noise are determined not by the barrier height, but instead by the barrier width. We further discuss consequences of simultaneous action of thermal and Lévy noises on statistics of passage times and population of reactants in double-well potentials.},
 bibtype = {article},
 author = {Capała, K. and Dybiec, B. and Gudowska-Nowak, E.},
 doi = {10.1063/1.5126263},
 journal = {Chaos},
 number = {1}
}

© 2020 Author(s). Stochastic evolution of various dynamic systems and reaction networks is commonly described in terms of noise assisted escape of an overdamped particle from a potential well, as devised by the paradigmatic Langevin equation in which additive Gaussian stochastic force reproduces effects of thermal fluctuations from the reservoir. When implemented for systems close to equilibrium, the approach correctly explains the emergence of the Boltzmann distribution for the ensemble of trajectories generated by the Langevin equation and relates the intensity of the noise strength to the mobility. This scenario can be further generalized to include effects of non-Gaussian, burstlike forcing modeled by Lévy noise. In this case, however, the pulsatile additive noise cannot be treated as the internal (thermal) since the relation between the strength of the friction and variance of the noise is violated. Heavy tails of Lévy noise distributions not only facilitate escape kinetics, but also, more importantly, change the escape protocol by altering the final stationary state to a non-Boltzmann, nonequilibrium form. As a result, contrary to the kinetics induced by a Gaussian white noise, escape rates in environments with Lévy noise are determined not by the barrier height, but instead by the barrier width. We further discuss consequences of simultaneous action of thermal and Lévy noises on statistics of passage times and population of reactants in double-well potentials.

@article{
 title = {First passage time moments of asymmetric Lévy flights},
 type = {article},
 year = {2020},
 keywords = {Lévy flight,first passage timemoments,fractional diffusion equation},
 volume = {53},
 id = {f77ef273-02fe-3a47-bea6-24fd2f72426c},
 created = {2020-10-30T10:12:17.495Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:17.495Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2020 The Author(s). Published by IOP Publishing Ltd. We investigate the first-passage dynamics of symmetric and asymmetric Lévy flights in semi-infinite and bounded intervals. By solving the space-fractional diffusion equation, we analyse the fractional-order moments of the first-passage time probability density function for different values of the index of stability and the skewness parameter. A comparison with results using the Langevin approach to Lévy flights is presented. For the semi-infinite domain, in certain special cases analytic results are derived explicitly, and in bounded intervals a general analytical expression for the mean first-passage time of Lévy flights with arbitrary skewness is presented. These results are complemented with extensive numerical analyses.},
 bibtype = {article},
 author = {Padash, A. and Padash, A. and Chechkin, A.V. and Chechkin, A.V. and Dybiec, B. and Magdziarz, M. and Shokri, B. and Shokri, B. and Metzler, R.},
 doi = {10.1088/1751-8121/ab9030},
 journal = {Journal of Physics A: Mathematical and Theoretical},
 number = {27}
}

© 2020 The Author(s). Published by IOP Publishing Ltd. We investigate the first-passage dynamics of symmetric and asymmetric Lévy flights in semi-infinite and bounded intervals. By solving the space-fractional diffusion equation, we analyse the fractional-order moments of the first-passage time probability density function for different values of the index of stability and the skewness parameter. A comparison with results using the Langevin approach to Lévy flights is presented. For the semi-infinite domain, in certain special cases analytic results are derived explicitly, and in bounded intervals a general analytical expression for the mean first-passage time of Lévy flights with arbitrary skewness is presented. These results are complemented with extensive numerical analyses.

@article{
 title = {Underdamped, anomalous kinetics in double-well potentials},
 type = {article},
 year = {2020},
 pages = {052123},
 volume = {102},
 websites = {https://link.aps.org/doi/10.1103/PhysRevE.102.052123},
 month = {11},
 day = {20},
 id = {fc06ed2c-3bc8-31d9-b1f7-56ddbbe9a44f},
 created = {2020-11-26T12:38:08.895Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-11-26T12:38:08.895Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 bibtype = {article},
 author = {Capała, Karol and Dybiec, Bartłomiej},
 doi = {10.1103/PhysRevE.102.052123},
 journal = {Physical Review E},
 number = {5}
}

@article{
 title = {Statistics of residence time for Lévy flights in unstable parabolic potentials},
 type = {article},
 year = {2020},
 pages = {042142},
 volume = {102},
 websites = {https://link.aps.org/doi/10.1103/PhysRevE.102.042142},
 month = {10},
 day = {29},
 id = {9f24ce88-2012-3c32-b59f-6ef043cfc88e},
 created = {2020-11-26T12:38:43.966Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-11-26T12:38:43.966Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 bibtype = {article},
 author = {Dubkov, Alexander A. and Dybiec, Bartłomiej and Spagnolo, Bernardo and Kharcheva, Anna and Guarcello, Claudio and Valenti, Davide},
 doi = {10.1103/PhysRevE.102.042142},
 journal = {Physical Review E},
 number = {4}
}

@article{
 title = {Lévy noise-driven escape from arctangent potential wells},
 type = {article},
 year = {2020},
 volume = {30},
 id = {b751898d-144e-3106-89d7-de071e6ad3e0},
 created = {2025-02-25T09:27:46.791Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2025-02-25T09:27:46.791Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {true},
 abstract = {The escape from a potential well is an archetypal problem in the study of stochastic dynamical systems, representing real-world situations from chemical reactions to leaving an established home range in movement ecology. Concurrently, Lévy noise is a well-established approach to model systems characterized by statistical outliers and diverging higher order moments, ranging from gene expression control to the movement patterns of animals and humans. Here, we study the problem of Lévy noise-driven escape from an almost rectangular, arctangent potential well restricted by two absorbing boundaries, mostly under the action of the Cauchy noise. We unveil analogies of the observed transient dynamics to the general properties of stationary states of Lévy processes in single-well potentials. The first-escape dynamics is shown to exhibit exponential tails. We examine the dependence of the escape on the shape parameters, steepness, and height of the arctangent potential. Finally, we explore in detail the behavior of the probability densities of the first-escape time and the last-hitting point. },
 bibtype = {article},
 author = {Capała, K. and Padash, A. and Chechkin, A.V. and Shokri, B. and Metzler, R. and Dybiec, B.},
 doi = {10.1063/5.0021795},
 journal = {Chaos},
 number = {12}
}

The escape from a potential well is an archetypal problem in the study of stochastic dynamical systems, representing real-world situations from chemical reactions to leaving an established home range in movement ecology. Concurrently, Lévy noise is a well-established approach to model systems characterized by statistical outliers and diverging higher order moments, ranging from gene expression control to the movement patterns of animals and humans. Here, we study the problem of Lévy noise-driven escape from an almost rectangular, arctangent potential well restricted by two absorbing boundaries, mostly under the action of the Cauchy noise. We unveil analogies of the observed transient dynamics to the general properties of stationary states of Lévy processes in single-well potentials. The first-escape dynamics is shown to exhibit exponential tails. We examine the dependence of the escape on the shape parameters, steepness, and height of the arctangent potential. Finally, we explore in detail the behavior of the probability densities of the first-escape time and the last-hitting point.

@article{
 title = {Deterministic and randomized motions in single-well potentials},
 type = {article},
 year = {2019},
 keywords = {Lévy walks,probability densities,randomized motion,single-well potential,velocity reversals},
 volume = {52},
 id = {72992e7d-31e7-30fe-bdfc-79b86da5de61},
 created = {2020-10-30T10:12:15.088Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.088Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2019 IOP Publishing Ltd. Newtonian, undamped motion in single-well potentials belong to a class of well-studied conservative systems. Here, we investigate and compare long-time properties of fully deterministic motions in single-well potentials with analogous randomized systems. We consider a special type of energyconserving randomization process: the deterministic motion is interrupted by hard velocity reversals v→(ti)→-v→ (ti) at random time instants ti. In the 1D case, for fixed initial conditions, the differences in probability distributions disappear in the long-time limit making asymptotic densities insensitive to the selection of random time instants when velocity is reversed. Substantially different probability distributions can be obtained, for instance, through the additional randomization of initial conditions. Analogously, in 2D setups, the probability distributions asymptotically are insensitive to a protocol of velocity reversals.},
 bibtype = {article},
 author = {Mandrysz, M. and Dybiec, B.},
 doi = {10.1088/1751-8121/ab4350},
 journal = {Journal of Physics A: Mathematical and Theoretical},
 number = {42}
}

© 2019 IOP Publishing Ltd. Newtonian, undamped motion in single-well potentials belong to a class of well-studied conservative systems. Here, we investigate and compare long-time properties of fully deterministic motions in single-well potentials with analogous randomized systems. We consider a special type of energyconserving randomization process: the deterministic motion is interrupted by hard velocity reversals v→(ti)→-v→ (ti) at random time instants ti. In the 1D case, for fixed initial conditions, the differences in probability distributions disappear in the long-time limit making asymptotic densities insensitive to the selection of random time instants when velocity is reversed. Substantially different probability distributions can be obtained, for instance, through the additional randomization of initial conditions. Analogously, in 2D setups, the probability distributions asymptotically are insensitive to a protocol of velocity reversals.

@article{
 title = {Stationary states for underdamped anharmonic oscillators driven by Cauchy noise},
 type = {article},
 year = {2019},
 volume = {29},
 id = {8fe50cfd-2744-3fd4-8d4e-2810c05f997a},
 created = {2020-10-30T10:12:15.148Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.148Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2019 Author(s). Using numerical methods, we have studied stationary states in the underdamped anharmonic stochastic oscillators driven by Cauchy noise. The shape of stationary states depends on both the potential type and the damping. If the damping is strong enough, for potential wells which in the overdamped regime produce multimodal stationary states, stationary states in the underdamped regime can be multimodal with the same number of modes like in the overdamped regime. For the parabolic potential, the stationary density is always unimodal, and it is given by the two dimensional α-stable density. For the mixture of quartic and parabolic single-well potentials, the stationary density can be bimodal. Nevertheless, the parabolic addition, which is strong enough, can destroy the bimodality of the stationary state.},
 bibtype = {article},
 author = {Capała, K. and Dybiec, B.},
 doi = {10.1063/1.5111637},
 journal = {Chaos},
 number = {9}
}

© 2019 Author(s). Using numerical methods, we have studied stationary states in the underdamped anharmonic stochastic oscillators driven by Cauchy noise. The shape of stationary states depends on both the potential type and the damping. If the damping is strong enough, for potential wells which in the overdamped regime produce multimodal stationary states, stationary states in the underdamped regime can be multimodal with the same number of modes like in the overdamped regime. For the parabolic potential, the stationary density is always unimodal, and it is given by the two dimensional α-stable density. For the mixture of quartic and parabolic single-well potentials, the stationary density can be bimodal. Nevertheless, the parabolic addition, which is strong enough, can destroy the bimodality of the stationary state.

@article{
 title = {Multimodal stationary states in symmetric single-well potentials driven by Cauchy noise},
 type = {article},
 year = {2019},
 keywords = {Brownian motion,numerical simulations,stationary states,stochastic particle dynamics},
 volume = {2019},
 id = {065a9938-925f-386b-8fd7-f08e0d046a43},
 created = {2020-10-30T10:12:15.175Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.175Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2019 IOP Publishing Ltd and SISSA Medialab srl. Stationary states for a particle moving in a single-well, steeper than parabolic, potential driven by Lévy noise can be bimodal. Here, we explore in details conditions that are required in order to induce multimodal stationary states having more than two modal values. Phenomenological arguments determining necessary conditions for emergence of stationary states of higher multimodality are provided. Basing on these arguments, appropriate symmetric single-well potentials are constructed. Finally, using numerical methods it is verified that stationary states have anticipated multimodality.},
 bibtype = {article},
 author = {Capała, K. and Dybiec, B.},
 doi = {10.1088/1742-5468/ab054c},
 journal = {Journal of Statistical Mechanics: Theory and Experiment},
 number = {3}
}

© 2019 IOP Publishing Ltd and SISSA Medialab srl. Stationary states for a particle moving in a single-well, steeper than parabolic, potential driven by Lévy noise can be bimodal. Here, we explore in details conditions that are required in order to induce multimodal stationary states having more than two modal values. Phenomenological arguments determining necessary conditions for emergence of stationary states of higher multimodality are provided. Basing on these arguments, appropriate symmetric single-well potentials are constructed. Finally, using numerical methods it is verified that stationary states have anticipated multimodality.

@article{
 title = {Energetics of single-well undamped stochastic oscillators},
 type = {article},
 year = {2019},
 volume = {99},
 id = {3468d41c-f9d2-35fa-9365-d87f630da615},
 created = {2020-10-30T10:12:15.212Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.212Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2019 American Physical Society. This paper discusses analytical and numerical results for nonharmonic, undamped, single-well, stochastic oscillators driven by additive noises. It focuses on average kinetic, potential, and total energies together with the corresponding distributions under random drivings, involving Gaussian white, Ornstein-Uhlenbeck, and Markovian dichotomous noises. It demonstrates that insensitivity of the average total energy to the single-well potential type, V(x) x2n, under Gaussian white noise does not extend to other noise types. Nevertheless, in the long-time limit (t→), the average energies grow as power law with exponents dependent on the steepness of the potential n. Another special limit corresponds to n→, i.e., to the infinite rectangular potential well, when the average total energy grows as a power law with the same exponent for all considered noise types.},
 bibtype = {article},
 author = {Mandrysz, M. and Dybiec, B.},
 doi = {10.1103/PhysRevE.99.012125},
 journal = {Physical Review E},
 number = {1}
}

© 2019 American Physical Society. This paper discusses analytical and numerical results for nonharmonic, undamped, single-well, stochastic oscillators driven by additive noises. It focuses on average kinetic, potential, and total energies together with the corresponding distributions under random drivings, involving Gaussian white, Ornstein-Uhlenbeck, and Markovian dichotomous noises. It demonstrates that insensitivity of the average total energy to the single-well potential type, V(x) x2n, under Gaussian white noise does not extend to other noise types. Nevertheless, in the long-time limit (t→), the average energies grow as power law with exponents dependent on the steepness of the potential n. Another special limit corresponds to n→, i.e., to the infinite rectangular potential well, when the average total energy grows as a power law with the same exponent for all considered noise types.

@article{
 title = {Multimodal stationary states under Cauchy noise},
 type = {article},
 year = {2019},
 volume = {99},
 id = {518c3fc1-1d6f-31b6-b4b9-d6ee43b14d20},
 created = {2020-10-30T10:12:16.207Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.207Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2019 American Physical Society. A Lévy noise is an efficient description of out-of-equilibrium systems. The presence of Lévy flights results in a plenitude of noise-induced phenomena. Among others, Lévy flights can produce stationary states with more than one modal value in single-well potentials. Here we explore stationary states in special double-well potentials demonstrating that a sufficiently high potential barrier separating potential wells can produce bimodal stationary states in each potential well. Furthermore, we explore how the decrease in the barrier height affects the multimodality of stationary states. Finally, we explore the role of multimodality of stationary states on noise-induced escape over the static potential barrier.},
 bibtype = {article},
 author = {Cieśla, M. and Capała, K. and Dybiec, B.},
 doi = {10.1103/PhysRevE.99.052118},
 journal = {Physical Review E},
 number = {5}
}

© 2019 American Physical Society. A Lévy noise is an efficient description of out-of-equilibrium systems. The presence of Lévy flights results in a plenitude of noise-induced phenomena. Among others, Lévy flights can produce stationary states with more than one modal value in single-well potentials. Here we explore stationary states in special double-well potentials demonstrating that a sufficiently high potential barrier separating potential wells can produce bimodal stationary states in each potential well. Furthermore, we explore how the decrease in the barrier height affects the multimodality of stationary states. Finally, we explore the role of multimodality of stationary states on noise-induced escape over the static potential barrier.

@article{
 title = {Entropy production and collective phenomena in biological channel gating},
 type = {article},
 year = {2019},
 volume = {50},
 id = {f8505664-8b1f-3067-8548-25dfae923b01},
 created = {2020-10-30T10:12:17.149Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:17.149Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2019 Jagellonian University. All rights reserved. We investigate gating kinetics of biological channels influenced by conformational changes within the membrane proteins forming the module, and subject to a coupling with other similar units. By introducing elements of stochastic thermodynamics, we analyze the information flow and associated entropy production during gating cycle of a single channel. In the second part of this paper, synchronized kinetics of multiple units of that type is analyzed in terms of Kuramoto's theory.},
 bibtype = {article},
 author = {Lisowski, B. and Zabicki, M. and Dybiec, B. and Gudowska-Nowak, E.},
 doi = {10.5506/APhysPolB.50.911},
 journal = {Acta Physica Polonica B},
 number = {5}
}

© 2019 Jagellonian University. All rights reserved. We investigate gating kinetics of biological channels influenced by conformational changes within the membrane proteins forming the module, and subject to a coupling with other similar units. By introducing elements of stochastic thermodynamics, we analyze the information flow and associated entropy production during gating cycle of a single channel. In the second part of this paper, synchronized kinetics of multiple units of that type is analyzed in terms of Kuramoto's theory.

@article{
 title = {Conservative random walks in confining potentials},
 type = {article},
 year = {2019},
 keywords = {Lévy flight,Lévy walk,conservative random walks},
 volume = {52},
 id = {791d3cc5-8ff9-3ad7-a331-333bb31bb017},
 created = {2020-10-30T10:12:17.193Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:17.193Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2018 IOP Publishing Ltd. Lévy walks are continuous time random walks with spatio-temporal coupling of jump lengths and waiting times, often used to model superdiffusive spreading processes such as animals searching for food, tracer motion in weakly chaotic systems, or even the dynamics in quantum systems such as cold atoms. In the simplest version Lévy walks move with a finite speed. Here, we present an extension of the Lévy walk scenario for the case when external force fields influence the motion. The resulting motion is a combination of the response to the deterministic force acting on the particle, changing its velocity according to the principle of total energy conservation, and random velocity reversals governed by the distribution of waiting times. For the fact that the motion stays conservative, that is, on a constant energy surface, our scenario is fundamentally different from thermal motion in the same external potentials. In particular, we present results for the velocity and position distributions for single well potentials of different steepness. The observed dynamics with its continuous velocity changes enriches the theory of Lévy walk processes and will be of use in a variety of systems, for which the particles are externally confined.},
 bibtype = {article},
 author = {Dybiec, B. and Capała, K. and Chechkin, A.V. and Metzler, R.},
 doi = {10.1088/1751-8121/aaefc2},
 journal = {Journal of Physics A: Mathematical and Theoretical},
 number = {1}
}

© 2018 IOP Publishing Ltd. Lévy walks are continuous time random walks with spatio-temporal coupling of jump lengths and waiting times, often used to model superdiffusive spreading processes such as animals searching for food, tracer motion in weakly chaotic systems, or even the dynamics in quantum systems such as cold atoms. In the simplest version Lévy walks move with a finite speed. Here, we present an extension of the Lévy walk scenario for the case when external force fields influence the motion. The resulting motion is a combination of the response to the deterministic force acting on the particle, changing its velocity according to the principle of total energy conservation, and random velocity reversals governed by the distribution of waiting times. For the fact that the motion stays conservative, that is, on a constant energy surface, our scenario is fundamentally different from thermal motion in the same external potentials. In particular, we present results for the velocity and position distributions for single well potentials of different steepness. The observed dynamics with its continuous velocity changes enriches the theory of Lévy walk processes and will be of use in a variety of systems, for which the particles are externally confined.

@article{
 title = {First-passage properties of asymmetric Lévy flights},
 type = {article},
 year = {2019},
 keywords = {Lévy flights,first-passage,search dynamics},
 volume = {52},
 id = {6a59ba37-671f-3783-a59d-f3596d8e619d},
 created = {2020-10-30T10:12:17.474Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:17.474Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2019 IOP Publishing Ltd. Lévy flights are paradigmatic generalised random walk processes, in which the independent stationary increments - the 'jump lengths' - are drawn from an α-stable jump length distribution with long-tailed, power-law asymptote. As a result, the variance of Lévy flights diverges and the trajectory is characterised by occasional extremely long jumps. Such long jumps significantly decrease the probability to revisit previous points of visitation, rendering Lévy flights efficient search processes in one and two dimensions. To further quantify their precise property as random search strategies we here study the first-passage time properties of Lévy flights in one-dimensional semi-infinite and bounded domains for symmetric and asymmetric jump length distributions. To obtain the full probability density function of first-passage times for these cases we employ two complementary methods. One approach is based on the space-fractional diffusion equation for the probability density function, from which the survival probability is obtained for different values of the stable index α and the skewness (asymmetry) parameter. The other approach is based on the stochastic Langevin equation with α-stable driving noise. Both methods have their advantages and disadvantages for explicit calculations and numerical evaluation, and the complementary approach involving both methods will be profitable for concrete applications. We also make use of the Skorokhod theorem for processes with independent increments and demonstrate that the numerical results are in good agreement with the analytical expressions for the probability density function of the first-passage times.},
 bibtype = {article},
 author = {Padash, A. and Chechkin, A.V. and Dybiec, B. and Pavlyukevich, I. and Shokri, B. and Metzler, R.},
 doi = {10.1088/1751-8121/ab493e},
 journal = {Journal of Physics A: Mathematical and Theoretical},
 number = {45}
}

© 2019 IOP Publishing Ltd. Lévy flights are paradigmatic generalised random walk processes, in which the independent stationary increments - the 'jump lengths' - are drawn from an α-stable jump length distribution with long-tailed, power-law asymptote. As a result, the variance of Lévy flights diverges and the trajectory is characterised by occasional extremely long jumps. Such long jumps significantly decrease the probability to revisit previous points of visitation, rendering Lévy flights efficient search processes in one and two dimensions. To further quantify their precise property as random search strategies we here study the first-passage time properties of Lévy flights in one-dimensional semi-infinite and bounded domains for symmetric and asymmetric jump length distributions. To obtain the full probability density function of first-passage times for these cases we employ two complementary methods. One approach is based on the space-fractional diffusion equation for the probability density function, from which the survival probability is obtained for different values of the stable index α and the skewness (asymmetry) parameter. The other approach is based on the stochastic Langevin equation with α-stable driving noise. Both methods have their advantages and disadvantages for explicit calculations and numerical evaluation, and the complementary approach involving both methods will be profitable for concrete applications. We also make use of the Skorokhod theorem for processes with independent increments and demonstrate that the numerical results are in good agreement with the analytical expressions for the probability density function of the first-passage times.

@article{
 title = {Energetics of the undamped stochastic harmonic oscillator},
 type = {article},
 year = {2018},
 volume = {49},
 id = {199d7538-5a84-3031-832d-7b0fa89d9621},
 created = {2020-10-30T10:12:15.243Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.243Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2018 Jagellonian University. All Rights Reserved. The harmonic oscillator is one of fundamental models in physics. In stochastic thermodynamics, such models are usually accompanied with both stochastic and damping forces, acting as energy counter-terms. Here, on the other hand, we study properties of the undamped harmonic oscillator driven by additive noises. Consequently, the popular cases of Gaussian white noise, Markovian dichotomous noise and Ornstein–Uhlenbeck noise are analyzed from the energy point of view employing both analytical and numerical methods. In accordance to one’s expectations, we confirm that energy is pumped into the system. We demonstrate that, as a function of time, initially total energy displays abrupt oscillatory changes, but then transits to the linear dependence in the long-time limit. Kinetic and potential parts of the energy are found to display oscillatory dependence at all times.},
 bibtype = {article},
 author = {Mandrysz, M. and Dybiec, B.},
 doi = {10.5506/APhysPolB.49.871},
 journal = {Acta Physica Polonica B},
 number = {5}
}

© 2018 Jagellonian University. All Rights Reserved. The harmonic oscillator is one of fundamental models in physics. In stochastic thermodynamics, such models are usually accompanied with both stochastic and damping forces, acting as energy counter-terms. Here, on the other hand, we study properties of the undamped harmonic oscillator driven by additive noises. Consequently, the popular cases of Gaussian white noise, Markovian dichotomous noise and Ornstein–Uhlenbeck noise are analyzed from the energy point of view employing both analytical and numerical methods. In accordance to one’s expectations, we confirm that energy is pumped into the system. We demonstrate that, as a function of time, initially total energy displays abrupt oscillatory changes, but then transits to the linear dependence in the long-time limit. Kinetic and potential parts of the energy are found to display oscillatory dependence at all times.

@article{
 title = {Thermodynamics of superdiffusion generated by Lévy-Wiener fluctuating forces},
 type = {article},
 year = {2018},
 keywords = {Fluctuation phenomena,Nonequilibrium and irreversible thermodynamics,Random walks and Lévy flights},
 volume = {20},
 id = {bf6a2f25-1c86-3dde-84bf-ed4b358f53a8},
 created = {2020-10-30T10:12:16.248Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.248Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2018 by the authors. Scale free Lévy motion is a generalized analogue of the Wiener process. Its time derivative extends the notion of "white noise" to non-Gaussian noise sources, and as such, it has been widely used to model natural signal variations described by an overdamped Langevin stochastic differential equation. Here, we consider the dynamics of an archetypal model: a Brownian-like particle is driven by external forces, and noise is represented by uncorrelated Lévy fluctuations. An unperturbed system of that form eventually attains a steady state which is uniquely determined by the set of parameter values. We show that the analyzed Markov process with the stability index α < 2 violates the detailed balance, i.e., its stationary state is quantified by a stationary probability density and nonvanishing current. We discuss consequences of the non-Gibbsian character of the stationary state of the system and its impact on the general form of the fluctuation-dissipation theorem derived for weak external forcing.},
 bibtype = {article},
 author = {Kuśmierz, Ł. and Dybiec, B. and Gudowska-Nowak, E.},
 doi = {10.3390/e20090658},
 journal = {Entropy},
 number = {9}
}

© 2018 by the authors. Scale free Lévy motion is a generalized analogue of the Wiener process. Its time derivative extends the notion of "white noise" to non-Gaussian noise sources, and as such, it has been widely used to model natural signal variations described by an overdamped Langevin stochastic differential equation. Here, we consider the dynamics of an archetypal model: a Brownian-like particle is driven by external forces, and noise is represented by uncorrelated Lévy fluctuations. An unperturbed system of that form eventually attains a steady state which is uniquely determined by the set of parameter values. We show that the analyzed Markov process with the stability index α < 2 violates the detailed balance, i.e., its stationary state is quantified by a stationary probability density and nonvanishing current. We discuss consequences of the non-Gibbsian character of the stationary state of the system and its impact on the general form of the fluctuation-dissipation theorem derived for weak external forcing.

@article{
 title = {Fighting for resources: Two leaders in the money addicted social hierarchies},
 type = {article},
 year = {2017},
 keywords = {Social dynamics,complex networks,hierarchy development,hierarchy maintenance},
 volume = {28},
 id = {1873f911-0875-3a5a-9cc8-c622e1c210d2},
 created = {2020-10-30T10:12:14.496Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:14.496Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2017 World Scientific Publishing Company. Building of hierarchy is inevitably associated with the constant competition for resources and attention. Here, we show how presence of two favored (leading) nodes affects properties of the network connecting individuals. In particular, we study how nodes characteristics depend on relative asymmetry between two leading nodes. It is shown that without strong and rigorous avoidance mechanism, individuals can support both dominating nodes. Slow redistribution of resources enhances this effect. Moreover, slow redistribution of resources results in development of social networks with a very limited number of layers.},
 bibtype = {article},
 author = {Dybiec, B.},
 doi = {10.1142/S0129183117500103},
 journal = {International Journal of Modern Physics C},
 number = {1}
}

© 2017 World Scientific Publishing Company. Building of hierarchy is inevitably associated with the constant competition for resources and attention. Here, we show how presence of two favored (leading) nodes affects properties of the network connecting individuals. In particular, we study how nodes characteristics depend on relative asymmetry between two leading nodes. It is shown that without strong and rigorous avoidance mechanism, individuals can support both dominating nodes. Slow redistribution of resources enhances this effect. Moreover, slow redistribution of resources results in development of social networks with a very limited number of layers.

@article{
 title = {Epidemics spread in heterogeneous populations},
 type = {article},
 year = {2017},
 keywords = {Computational Methods},
 volume = {90},
 id = {81747b9c-c800-3c52-ade1-b9a49ecae687},
 created = {2020-10-30T10:12:15.303Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.303Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2017, The Author(s). Individuals building populations are subject to variability. This variability affects progress of epidemic outbreaks, because individuals tend to be more or less resistant. Individuals also differ with respect to their recovery rate. Here, properties of the SIR model in inhomogeneous populations are studied. It is shown that a small change in model’s parameters, e.g. recovery or infection rate, can substantially change properties of final states which is especially well-visible in distributions of the epidemic size. In addition to the epidemic size and radii distributions, the paper explores first passage time properties of epidemic outbreaks.},
 bibtype = {article},
 author = {Capała, K. and Dybiec, B.},
 doi = {10.1140/epjb/e2017-70723-6},
 journal = {European Physical Journal B},
 number = {5}
}

© 2017, The Author(s). Individuals building populations are subject to variability. This variability affects progress of epidemic outbreaks, because individuals tend to be more or less resistant. Individuals also differ with respect to their recovery rate. Here, properties of the SIR model in inhomogeneous populations are studied. It is shown that a small change in model’s parameters, e.g. recovery or infection rate, can substantially change properties of final states which is especially well-visible in distributions of the epidemic size. In addition to the epidemic size and radii distributions, the paper explores first passage time properties of epidemic outbreaks.

@article{
 title = {Underdamped stochastic harmonic oscillator driven by Lévy noise},
 type = {article},
 year = {2017},
 volume = {96},
 id = {43822ea3-a6d2-39b7-8170-5e820bbc4a2b},
 created = {2020-10-30T10:12:16.327Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.327Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2017 American Physical Society. We investigate the distribution of potential and kinetic energy in stationary states of the linearly damped stochastic oscillator driven by Lévy noises. In the long time limit distributions of kinetic and potential energies of the oscillator follow the power-law asymptotics and do not fulfill the equipartition theorem. The partition of the mechanical energy is controlled by the damping coefficient. In the limit of vanishing damping a stochastic analog of the equipartition theorem can be proposed, namely, the statistical properties of potential and kinetic energies attain distributions characterized by the same widths. For larger damping coefficient the larger fraction of energy is stored in its potential form. In the limit of very strong damping the contribution of kinetic energy becomes negligible. Finally, we demonstrate that the ratio of instantaneous kinetic and potential energies, which signifies departure from the mechanical energy equipartition, follows universal power-law asymptotics, regardless of the symmetric α-stable noise parameters. Altogether our investigations clearly indicate strongly nonequilibrium character of Lévy-stable fluctuations with the stability index α<2.},
 bibtype = {article},
 author = {Dybiec, B. and Gudowska-Nowak, E. and Sokolov, I.M.},
 doi = {10.1103/PhysRevE.96.042118},
 journal = {Physical Review E},
 number = {4}
}

© 2017 American Physical Society. We investigate the distribution of potential and kinetic energy in stationary states of the linearly damped stochastic oscillator driven by Lévy noises. In the long time limit distributions of kinetic and potential energies of the oscillator follow the power-law asymptotics and do not fulfill the equipartition theorem. The partition of the mechanical energy is controlled by the damping coefficient. In the limit of vanishing damping a stochastic analog of the equipartition theorem can be proposed, namely, the statistical properties of potential and kinetic energies attain distributions characterized by the same widths. For larger damping coefficient the larger fraction of energy is stored in its potential form. In the limit of very strong damping the contribution of kinetic energy becomes negligible. Finally, we demonstrate that the ratio of instantaneous kinetic and potential energies, which signifies departure from the mechanical energy equipartition, follows universal power-law asymptotics, regardless of the symmetric α-stable noise parameters. Altogether our investigations clearly indicate strongly nonequilibrium character of Lévy-stable fluctuations with the stability index α<2.

@article{
 title = {Lévy flights versus Lévy walks in bounded domains},
 type = {article},
 year = {2017},
 volume = {95},
 id = {b794f7b5-6848-3d02-9abb-851cfa9db939},
 created = {2020-10-30T10:12:17.209Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:17.209Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2017 American Physical Society. Lévy flights and Lévy walks serve as two paradigms of random walks resembling common features but also bearing fundamental differences. One of the main dissimilarities is the discontinuity versus continuity of their trajectories and infinite versus finite propagation velocity. As a consequence, a well-developed theory of Lévy flights is associated with their pathological physical properties, which in turn are resolved by the concept of Lévy walks. Here, we explore Lévy flight and Lévy walk models on bounded domains, examining their differences and analogies. We investigate analytically and numerically whether and under which conditions both approaches yield similar results in terms of selected statistical observables characterizing the motion: the survival probability, mean first passage time, and stationary probability density functions. It is demonstrated that the similarity of the models is affected by the type of boundary conditions and the value of the stability index defining the asymptotics of the jump length distribution.},
 bibtype = {article},
 author = {Dybiec, B. and Gudowska-Nowak, E. and Barkai, E. and Dubkov, A.A.},
 doi = {10.1103/PhysRevE.95.052102},
 journal = {Physical Review E},
 number = {5}
}

© 2017 American Physical Society. Lévy flights and Lévy walks serve as two paradigms of random walks resembling common features but also bearing fundamental differences. One of the main dissimilarities is the discontinuity versus continuity of their trajectories and infinite versus finite propagation velocity. As a consequence, a well-developed theory of Lévy flights is associated with their pathological physical properties, which in turn are resolved by the concept of Lévy walks. Here, we explore Lévy flight and Lévy walk models on bounded domains, examining their differences and analogies. We investigate analytically and numerically whether and under which conditions both approaches yield similar results in terms of selected statistical observables characterizing the motion: the survival probability, mean first passage time, and stationary probability density functions. It is demonstrated that the similarity of the models is affected by the type of boundary conditions and the value of the stability index defining the asymptotics of the jump length distribution.

@article{
 title = {Escape from a disk with partly absorbing boundaries driven by bi-variate α-stable noises},
 type = {article},
 year = {2016},
 keywords = {diffusion,fluctuations (theory),stochastic particle dynamics (theory)},
 volume = {2016},
 id = {7e2c5520-3d38-31a9-a910-bdcca2ae32b9},
 created = {2020-10-30T10:12:15.366Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.366Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2016 IOP Publishing Ltd and SISSA Medialab srl. In this paper we explore the problem of first escape from the disk. The part of the disk edge, i.e. the arc controlled by the angle φ, is absorbing. For an appropriate choice of parameters the mean first passage time (MFPT) is a non-monotonous function of the stability index α describing jumps' length asymptotics. The dependence of the MFPT on the stability index α is sensitive to the way in which the non-absorbing part of the disk edge is accounted for.},
 bibtype = {article},
 author = {Szczepaniec, K. and Dybiec, B.},
 doi = {10.1088/1742-5468/2016/05/054005},
 journal = {Journal of Statistical Mechanics: Theory and Experiment},
 number = {5}
}

© 2016 IOP Publishing Ltd and SISSA Medialab srl. In this paper we explore the problem of first escape from the disk. The part of the disk edge, i.e. the arc controlled by the angle φ, is absorbing. For an appropriate choice of parameters the mean first passage time (MFPT) is a non-monotonous function of the stability index α describing jumps' length asymptotics. The dependence of the MFPT on the stability index α is sensitive to the way in which the non-absorbing part of the disk edge is accounted for.

@article{
 title = {To hit or to pass it over - Remarkable transient behavior of first arrivals and passages for Lévy flights in finite domains},
 type = {article},
 year = {2016},
 keywords = {Lévy flights,first arrival,first escape,leapovers},
 volume = {49},
 id = {427fe394-8c09-3bbc-8da0-79deab548aba},
 created = {2020-10-30T10:12:16.352Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.352Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2016 IOP Publishing Ltd. The term 'Lévy flights' was coined by Benoit Mandelbrot, who thus poeticized α-stable Lévy random motion, a Markovian process with stationary independent increments distributed according to the α-stable Lévy probability law. Contrary to the Brownian motion, the trajectories of the α-stable Lévy motion are discontinous, that is exhibit jumps. This feature implies that the process of first passage through the boundary of a given space domain, or the first escape, is different from the process of first arrival (hit) at the boundary. Here we investigate the properties of first escapes and first arrivals for Lévy flights and explore how the asymptotic behavior of the corresponding (passage and hit) probabilities is sensitive to the size of the domain. In particular, we find that the survival probability to stay in a large enough, finite domain has a universal Sparre Andersen temporal scaling , which is transient and changes to an exponential non-universal decay at longer times. Also, the probability to arrive at a finite domain possesses a similar transient Sparre Andersen universality that turns into a non-universal and slower power-law decay in course of time. Finally, we demonstrate that the probability density of the leapover length ℓ over the boundary, related to overshooting events, has an intermediate asymptotics () which is inherent for the escape from a semi-infinite domain. However, for larger leapovers the probability density decays faster according to the law. Thus, we find that the laws derived for the α-stable processes on the semi-infinite domain, manifest themselves as transients for Lévy flights on the finite domain.},
 bibtype = {article},
 author = {Dybiec, B. and Gudowska-Nowak, E. and Chechkin, A.},
 doi = {10.1088/1751-8113/49/50/504001},
 journal = {Journal of Physics A: Mathematical and Theoretical},
 number = {50}
}

© 2016 IOP Publishing Ltd. The term 'Lévy flights' was coined by Benoit Mandelbrot, who thus poeticized α-stable Lévy random motion, a Markovian process with stationary independent increments distributed according to the α-stable Lévy probability law. Contrary to the Brownian motion, the trajectories of the α-stable Lévy motion are discontinous, that is exhibit jumps. This feature implies that the process of first passage through the boundary of a given space domain, or the first escape, is different from the process of first arrival (hit) at the boundary. Here we investigate the properties of first escapes and first arrivals for Lévy flights and explore how the asymptotic behavior of the corresponding (passage and hit) probabilities is sensitive to the size of the domain. In particular, we find that the survival probability to stay in a large enough, finite domain has a universal Sparre Andersen temporal scaling , which is transient and changes to an exponential non-universal decay at longer times. Also, the probability to arrive at a finite domain possesses a similar transient Sparre Andersen universality that turns into a non-universal and slower power-law decay in course of time. Finally, we demonstrate that the probability density of the leapover length ℓ over the boundary, related to overshooting events, has an intermediate asymptotics () which is inherent for the escape from a semi-infinite domain. However, for larger leapovers the probability density decays faster according to the law. Thus, we find that the laws derived for the α-stable processes on the semi-infinite domain, manifest themselves as transients for Lévy flights on the finite domain.

@article{
 title = {Non-equilibrium escape problems under bivariate alpha;-stable noises},
 type = {article},
 year = {2016},
 volume = {47},
 id = {4b2119fc-5a31-324f-9a84-1d19229a7c1b},
 created = {2020-10-30T10:12:17.245Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:17.245Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {Stochastic resonance is a prominent effect consisting in enhancement of a response of a physical system to deterministic driving in the presence of noise. It demonstrates a constructive role the noise may play in increasing the sensitivity of the system to weak signals, and emerges in different theoretical models and experimental situations. We consider this effect in a periodically modulated two-dimensional double-well potential under the influence of an isotropic alpha;-stable noise, and discuss the performance of various measures used to describe the stochastic resonance in other setups.},
 bibtype = {article},
 author = {Dybiec, B. and Szczepaniec, K. and Kac, M. and Sokolov, I.M.},
 doi = {10.5506/APhysPolB.47.1327},
 journal = {Acta Physica Polonica B},
 number = {5}
}

Stochastic resonance is a prominent effect consisting in enhancement of a response of a physical system to deterministic driving in the presence of noise. It demonstrates a constructive role the noise may play in increasing the sensitivity of the system to weak signals, and emerges in different theoretical models and experimental situations. We consider this effect in a periodically modulated two-dimensional double-well potential under the influence of an isotropic alpha;-stable noise, and discuss the performance of various measures used to describe the stochastic resonance in other setups.

@article{
 title = {Spectral characteristics of steady-state Lévy flights in confinement potential profiles},
 type = {article},
 year = {2016},
 keywords = {rigorous results in statistical mechanics,stochastic particle dynamics,stochastic processes (theory)},
 volume = {2016},
 id = {c9f3f4c1-e3b7-3e6e-891c-92284ddefac9},
 created = {2020-10-30T10:12:17.430Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:17.430Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2016 IOP Publishing Ltd and SISSA Medialab srl. The steady-state correlation characteristics of superdiffusion in the form of Levy flights in one-dimensional confinement potential profiles are investigated both theoretically and numerically. Specifically, for Cauchy stable noise we calculate the steady-state probability density function for an infinitely deep rectangular potential well and for a symmetric steep potential well of the type U(x)∞x2m. For these potential profiles and arbitrary Levy index α, we obtain the asymptotic expression of the spectral power density.},
 bibtype = {article},
 author = {Kharcheva, A.A. and Dubkov, A.A. and Dybiec, B. and Spagnolo, B. and Valenti, D.},
 doi = {10.1088/1742-5468/2016/05/054039},
 journal = {Journal of Statistical Mechanics: Theory and Experiment},
 number = {5}
}

© 2016 IOP Publishing Ltd and SISSA Medialab srl. The steady-state correlation characteristics of superdiffusion in the form of Levy flights in one-dimensional confinement potential profiles are investigated both theoretically and numerically. Specifically, for Cauchy stable noise we calculate the steady-state probability density function for an infinitely deep rectangular potential well and for a symmetric steep potential well of the type U(x)∞x2m. For these potential profiles and arbitrary Levy index α, we obtain the asymptotic expression of the spectral power density.

@article{
 title = {Escape from hypercube driven by multi-variate α-stable noises: role of independence},
 type = {article},
 year = {2015},
 keywords = {Statistical and Nonlinear Physics},
 volume = {88},
 id = {82cf3e1c-525f-3949-9629-66e5289731bf},
 created = {2020-10-30T10:12:15.437Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.437Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2015, The Author(s). We explore properties of the escape kinetics from the d-dimensional hypercube driven by multi-variate α-stable noises. Using methods of stochastic dynamics we show complex dependence of the mean first passage time for the escape from the hypercube as a function of the hypercube dimension d. Finally, we show how the escape process can be used to quantify independence of components of multi-variate α-stable noises.},
 bibtype = {article},
 author = {Dybiec, B. and Szczepaniec, K.},
 doi = {10.1140/epjb/e2015-60429-2},
 journal = {European Physical Journal B},
 number = {7}
}

© 2015, The Author(s). We explore properties of the escape kinetics from the d-dimensional hypercube driven by multi-variate α-stable noises. Using methods of stochastic dynamics we show complex dependence of the mean first passage time for the escape from the hypercube as a function of the hypercube dimension d. Finally, we show how the escape process can be used to quantify independence of components of multi-variate α-stable noises.

@article{
 title = {Escape from bounded domains driven by multivariate -stable noises},
 type = {article},
 year = {2015},
 keywords = {diffusion,driven diffusive systems (theory),stochastic particle dynamics (theory),stochastic processes (theory)},
 volume = {2015},
 id = {56bbb732-4b22-3530-95a0-31b1cb956c54},
 created = {2020-10-30T10:12:15.452Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.452Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2015 IOP Publishing Ltd and SISSA Medialab srl. In this paper we provide an analysis of a mean first passage time problem of a random walker subject to a bivariate -stable Lévy-type noise from a 2-dimensional disk. For an appropriate choice of parameters the mean first passage time reveals non-trivial, non-monotonous dependence on the stability indexdescribing jumps' length asymptotics both for spherical and Cartesian Lévy flights. Finally, we study escape from a d-dimensional hypersphere showing that a d-dimensional escape process can be used to discriminate between various types of multivariate -stable noises, especially spherical and Cartesian Lévy flights.},
 bibtype = {article},
 author = {Szczepaniec, K. and Dybiec, B.},
 doi = {10.1088/1742-5468/2015/06/P06031},
 journal = {Journal of Statistical Mechanics: Theory and Experiment},
 number = {6}
}

© 2015 IOP Publishing Ltd and SISSA Medialab srl. In this paper we provide an analysis of a mean first passage time problem of a random walker subject to a bivariate -stable Lévy-type noise from a 2-dimensional disk. For an appropriate choice of parameters the mean first passage time reveals non-trivial, non-monotonous dependence on the stability indexdescribing jumps' length asymptotics both for spherical and Cartesian Lévy flights. Finally, we study escape from a d-dimensional hypersphere showing that a d-dimensional escape process can be used to discriminate between various types of multivariate -stable noises, especially spherical and Cartesian Lévy flights.

@article{
 title = {Estimation of the smallest eigenvalue in fractional escape problems: Semi-analytics and fits},
 type = {article},
 year = {2015},
 keywords = {Mittag-Leffler distribution,Nonlinear curve fitting,Survival probability},
 volume = {187},
 id = {da58adf2-527f-324e-8e1b-950b994ea40e},
 created = {2020-10-30T10:12:15.487Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.487Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2014 Elsevier B.V. All rights reserved. Continuous time random walks with heavy tailed distributions of waiting times and jump lengths lead to situations when evolution of a probability density of finding a particle at given point at given time is described by the bi-fractional Smoluchowski-Fokker-Planck equation. A power-law distribution of waiting times results in very general properties of a survival probability which in turn can be used to estimate eigenvalues of some fractional operators. Here, the problem of numerical estimation of the smallest eigenvalues is discussed for the two generic problems: escape from a finite interval and the Kramers problem of escape from a potential well. We discuss both how to numerically obtain the (effective) smallest eigenvalue of the problem, and how it can be used in numerically assessing other important characteristics of the processes.},
 bibtype = {article},
 author = {Dybiec, B. and Sokolov, I.M.},
 doi = {10.1016/j.cpc.2014.10.007},
 journal = {Computer Physics Communications}
}

© 2014 Elsevier B.V. All rights reserved. Continuous time random walks with heavy tailed distributions of waiting times and jump lengths lead to situations when evolution of a probability density of finding a particle at given point at given time is described by the bi-fractional Smoluchowski-Fokker-Planck equation. A power-law distribution of waiting times results in very general properties of a survival probability which in turn can be used to estimate eigenvalues of some fractional operators. Here, the problem of numerical estimation of the smallest eigenvalues is discussed for the two generic problems: escape from a finite interval and the Kramers problem of escape from a potential well. We discuss both how to numerically obtain the (effective) smallest eigenvalue of the problem, and how it can be used in numerically assessing other important characteristics of the processes.

@article{
 title = {Taming Lévy flights in confined crowded geometries},
 type = {article},
 year = {2015},
 volume = {142},
 month = {4},
 publisher = {American Institute of Physics Inc.},
 day = {28},
 id = {49db0460-c0bf-3bc5-9929-ef67d642d4d2},
 created = {2020-10-30T10:12:17.269Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-11-02T08:55:05.701Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2015 AIP Publishing LLC. We study two-dimensional diffusive motion of a tracer particle in restricted, crowded anisotropic geometries. The underlying medium is formed from a monolayer of elongated molecules [Cieśla J. Chem. Phys. 140, 044706 (2014)] of known concentration. Within this mesh structure, a tracer molecule is allowed to perform a Cauchy random walk with uncorrelated steps. Our analysis shows that the presence of obstacles significantly influences the motion, which in an obstacle-free space would be of a superdiffusive type. At the same time, the selfdiffusive process reveals different anomalous properties, both at the level of a single trajectory realization and after the ensemble averaging. In particular, due to obstacles, the sample mean squared displacement asymptotically grows sublinearly in time, suggesting a non-Markov character of motion. Closer inspection of survival probabilities indicates, however, that the underlying diffusion is memoryless over long time scales despite a strong inhomogeneity of the motion induced by the orientational ordering.},
 bibtype = {article},
 author = {Cieśla, M. and Dybiec, B. and Sokolov, I. and Gudowska-Nowak, E.},
 doi = {10.1063/1.4919368},
 journal = {Journal of Chemical Physics},
 number = {16}
}

© 2015 AIP Publishing LLC. We study two-dimensional diffusive motion of a tracer particle in restricted, crowded anisotropic geometries. The underlying medium is formed from a monolayer of elongated molecules [Cieśla J. Chem. Phys. 140, 044706 (2014)] of known concentration. Within this mesh structure, a tracer molecule is allowed to perform a Cauchy random walk with uncorrelated steps. Our analysis shows that the presence of obstacles significantly influences the motion, which in an obstacle-free space would be of a superdiffusive type. At the same time, the selfdiffusive process reveals different anomalous properties, both at the level of a single trajectory realization and after the ensemble averaging. In particular, due to obstacles, the sample mean squared displacement asymptotically grows sublinearly in time, suggesting a non-Markov character of motion. Closer inspection of survival probabilities indicates, however, that the underlying diffusion is memoryless over long time scales despite a strong inhomogeneity of the motion induced by the orientational ordering.

@article{
 title = {Escape from finite intervals: Numerical studies of order statistics},
 type = {article},
 year = {2014},
 volume = {45},
 id = {09fe021d-052e-3701-80d9-569a3068f4da},
 created = {2020-10-30T10:12:14.496Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:14.496Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {The subdiffusive systems are characterized by the diverging mean residence time. The escape of a subdiffusive particle from finite intervals cannot be characterized by the mean exit time. The situation significantly changes when instead of a single subdiffusive particle there is an ensemble of subdiffusive particles. In such a case, if the ensemble of particles is large enough, the mean minimal first escape time (first exit time of the fastest particle) is well defined quantity and the minimal first exit time distribution has fast decaying power-law asymptotics. Consequently, the increase in the number of particles facilitates escape kinetics and shortenes the system's lifetime.},
 bibtype = {article},
 author = {Dybiec, B.},
 doi = {10.5506/APhysPolB.45.1037},
 journal = {Acta Physica Polonica B},
 number = {5}
}

The subdiffusive systems are characterized by the diverging mean residence time. The escape of a subdiffusive particle from finite intervals cannot be characterized by the mean exit time. The situation significantly changes when instead of a single subdiffusive particle there is an ensemble of subdiffusive particles. In such a case, if the ensemble of particles is large enough, the mean minimal first escape time (first exit time of the fastest particle) is well defined quantity and the minimal first exit time distribution has fast decaying power-law asymptotics. Consequently, the increase in the number of particles facilitates escape kinetics and shortenes the system's lifetime.

@article{
 title = {Resonant activation in 2D and 3D systems driven by multi-variate Lévy noise},
 type = {article},
 year = {2014},
 keywords = {diffusion,stochastic particle dynamics (theory),stochastic processes (theory)},
 volume = {2014},
 id = {dd194406-7516-3a9b-ab25-6a68211f83c8},
 created = {2020-10-30T10:12:15.515Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.515Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2014 IOP Publishing Ltd and SISSA Medialab srl. Resonant activation is one of the classical effects demonstrating the constructive role of noise. In resonant activation, the cooperative action of a barrier modulation process and noise lead to the optimal escape kinetics as measured by the mean first passage time. Resonant activation has been observed in versatile systems for various types of barrier modulation process and noise type. Here, we show that resonant activation is also observed in 2D and 3D systems driven by bi-variate and tri-variate α-stable noise. The strength of resonant activation is sensitive to the exact value of the noise parameters. In particular, the decrease in the stability index α results in the disappearance of the resonant activation.},
 bibtype = {article},
 author = {Szczepaniec, K. and Dybiec, B.},
 doi = {10.1088/1742-5468/2014/09/P09022},
 journal = {Journal of Statistical Mechanics: Theory and Experiment},
 number = {9}
}

© 2014 IOP Publishing Ltd and SISSA Medialab srl. Resonant activation is one of the classical effects demonstrating the constructive role of noise. In resonant activation, the cooperative action of a barrier modulation process and noise lead to the optimal escape kinetics as measured by the mean first passage time. Resonant activation has been observed in versatile systems for various types of barrier modulation process and noise type. Here, we show that resonant activation is also observed in 2D and 3D systems driven by bi-variate and tri-variate α-stable noise. The strength of resonant activation is sensitive to the exact value of the noise parameters. In particular, the decrease in the stability index α results in the disappearance of the resonant activation.

@article{
 title = {Stationary states in two-dimensional systems driven by bivariate Lévy noises},
 type = {article},
 year = {2014},
 volume = {90},
 id = {e5a62e05-723d-3055-9607-72a8f5f6771f},
 created = {2020-10-30T10:12:15.540Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.540Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2014 American Physical Society. Systems driven by α-stable noises could be very different from their Gaussian counterparts. Stationary states in single-well potentials can be multimodal. Moreover, a potential well needs to be steep enough in order to produce stationary states. Here it is demonstrated that two-dimensional (2D) systems driven by bivariate α-stable noises are even more surprising than their 1D analogs. In 2D systems, intriguing properties of stationary states originate not only due to heavy tails of noise pulses, which are distributed according to α-stable densities, but also because of properties of spectral measures. Consequently, 2D systems are described by a whole family of Langevin and fractional diffusion equations. Solutions of these equations bear some common properties, but also can be very different. It is demonstrated that also for 2D systems potential wells need to be steep enough in order to produce bounded states. Moreover, stationary states can have local minima at the origin. The shape of stationary states reflects symmetries of the underlying noise, i.e., its spectral measure. Finally, marginal densities in power-law potentials also have power-law asymptotics.},
 bibtype = {article},
 author = {Szczepaniec, K. and Dybiec, B.},
 doi = {10.1103/PhysRevE.90.032128},
 journal = {Physical Review E - Statistical, Nonlinear, and Soft Matter Physics},
 number = {3}
}

© 2014 American Physical Society. Systems driven by α-stable noises could be very different from their Gaussian counterparts. Stationary states in single-well potentials can be multimodal. Moreover, a potential well needs to be steep enough in order to produce stationary states. Here it is demonstrated that two-dimensional (2D) systems driven by bivariate α-stable noises are even more surprising than their 1D analogs. In 2D systems, intriguing properties of stationary states originate not only due to heavy tails of noise pulses, which are distributed according to α-stable densities, but also because of properties of spectral measures. Consequently, 2D systems are described by a whole family of Langevin and fractional diffusion equations. Solutions of these equations bear some common properties, but also can be very different. It is demonstrated that also for 2D systems potential wells need to be steep enough in order to produce bounded states. Moreover, stationary states can have local minima at the origin. The shape of stationary states reflects symmetries of the underlying noise, i.e., its spectral measure. Finally, marginal densities in power-law potentials also have power-law asymptotics.

@article{
 title = {Quantifying a resonant-activation-like phenomenon in non-Markovian systems},
 type = {article},
 year = {2014},
 volume = {89},
 id = {fd411413-029a-3a8f-a143-de6a4656a300},
 created = {2020-10-30T10:12:15.580Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.580Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {Resonant activation is an effect of a noise-induced escape over a modulated potential barrier. The modulation of an energy landscape facilitates the escape kinetics and makes it optimal as measured by the mean first-passage time. A canonical example of resonant activation is a Brownian particle moving in a time-dependent potential under action of Gaussian white noise. Resonant activation is observed not only in typical Markovian-Gaussian systems but also in far-from-equilibrium and far-from-Markovianity regimes. We demonstrate that using an alternative to the mean first-passage time, robust measures of resonant activation, the signature of this effect can be observed in general continuous-time random walks in modulated potentials, even in situations when the mean first-passage time diverges. © 2014 American Physical Society.},
 bibtype = {article},
 author = {Szczepaniec, K. and Dybiec, B.},
 doi = {10.1103/PhysRevE.89.042138},
 journal = {Physical Review E - Statistical, Nonlinear, and Soft Matter Physics},
 number = {4}
}

Resonant activation is an effect of a noise-induced escape over a modulated potential barrier. The modulation of an energy landscape facilitates the escape kinetics and makes it optimal as measured by the mean first-passage time. A canonical example of resonant activation is a Brownian particle moving in a time-dependent potential under action of Gaussian white noise. Resonant activation is observed not only in typical Markovian-Gaussian systems but also in far-from-equilibrium and far-from-Markovianity regimes. We demonstrate that using an alternative to the mean first-passage time, robust measures of resonant activation, the signature of this effect can be observed in general continuous-time random walks in modulated potentials, even in situations when the mean first-passage time diverges. © 2014 American Physical Society.

@article{
 title = {Modeling rises and falls in money addicted social hierarchies},
 type = {article},
 year = {2014},
 keywords = {entropy and other measures of information,fluctuation phenomena,random processes,social and economic systems},
 volume = {89},
 id = {9634269d-3302-32f6-9a34-76cb14306da4},
 created = {2020-10-30T10:12:16.407Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.407Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {The emergence of large communities is inherently associated with the creation of social structures. Connections between individuals are indispensable for cooperative action of agents building social groups. Moreover, social groups usually evolve and their structure changes over time. Consequently, an underlying network connecting individuals is not static, reflecting an ongoing adaptation to new conditions. The evolution of social connections is influenced by the relative position (hierarchy) of individuals building the system as well as by the availability of resources. We explore this aspect of human ambition by modeling the interplay of social networking and an uneven distribution of external resources. The model naturally generates social hierarchies. Remarkably, this social structure exhibits a rise-and-fall behavior. A well pronounced quasi-periodic dynamics, which is closely associated with the dissipation of resources that are needed to sustain the social links, is revealed. © 2014 The Royal Swedish Academy of Sciences.},
 bibtype = {article},
 author = {Dybiec, B. and Mitarai, N. and Sneppen, K.},
 doi = {10.1088/0031-8949/89/8/085002},
 journal = {Physica Scripta},
 number = {8}
}

The emergence of large communities is inherently associated with the creation of social structures. Connections between individuals are indispensable for cooperative action of agents building social groups. Moreover, social groups usually evolve and their structure changes over time. Consequently, an underlying network connecting individuals is not static, reflecting an ongoing adaptation to new conditions. The evolution of social connections is influenced by the relative position (hierarchy) of individuals building the system as well as by the availability of resources. We explore this aspect of human ambition by modeling the interplay of social networking and an uneven distribution of external resources. The model naturally generates social hierarchies. Remarkably, this social structure exhibits a rise-and-fall behavior. A well pronounced quasi-periodic dynamics, which is closely associated with the dissipation of resources that are needed to sustain the social links, is revealed. © 2014 The Royal Swedish Academy of Sciences.

@article{
 title = {Improving epidemic control strategies by extended detection},
 type = {article},
 year = {2014},
 keywords = {Epidemiological modeling,disease spread,epidemiological control,stochastic modeling},
 volume = {25},
 id = {b7aad5c3-7f8f-346e-9c67-f958bd03050e},
 created = {2020-10-30T10:12:16.421Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.421Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {Majority of epidemics eradication programs work in preventive responsive way. The lack of exact information about epidemiological status of individuals makes responsive actions less efficient. Here, we demonstrate that additional tests can significantly increase the efficiency of blind treatment (vaccination or culling). Eradication strategy consisting of blind treatment in very limited local neighborhood supplemented by extra tests in a little bit larger neighborhood is able to prevent invasion of even highly infectious diseases and to achieve this at a cost lower than for the blind strategy. The effectiveness of the extended strategy depends on such parameters as the test efficiency and test cost. © 2014 World Scientific Publishing Company.},
 bibtype = {article},
 author = {Karp, P. and Dybiec, B. and Kleczkowski, A.},
 doi = {10.1142/S0129183113501064},
 journal = {International Journal of Modern Physics C},
 number = {4}
}

Majority of epidemics eradication programs work in preventive responsive way. The lack of exact information about epidemiological status of individuals makes responsive actions less efficient. Here, we demonstrate that additional tests can significantly increase the efficiency of blind treatment (vaccination or culling). Eradication strategy consisting of blind treatment in very limited local neighborhood supplemented by extra tests in a little bit larger neighborhood is able to prevent invasion of even highly infectious diseases and to achieve this at a cost lower than for the blind strategy. The effectiveness of the extended strategy depends on such parameters as the test efficiency and test cost. © 2014 World Scientific Publishing Company.

@book{
 title = {Mittag-leffler pattern in anomalous diffusion},
 type = {book},
 year = {2013},
 source = {Lecture Notes in Electrical Engineering},
 keywords = {Mittag-Leffler function,anomalous diffusion,bi-fractional Fokker-Planck-Smoluchowski equation},
 volume = {257 LNEE},
 id = {a7ed4b25-9509-3365-81fe-592e750db80f},
 created = {2020-10-30T10:12:14.566Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:14.566Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {Various systems described by the bi-fractional Fokker-Planck-Smoluchowski equation display some very general and universal properties. These universal characteristics originate in the underlying competition between long jumps (fractional space derivative) and long waiting times (fractional time derivative). Using a few selected model examples the universal features of anomalous diffusion will be demonstrated. © 2013 Springer International Publishing Switzerland.},
 bibtype = {book},
 author = {Dybiec, B.},
 doi = {10.1007/978-3-319-00933-9-12}
}

Various systems described by the bi-fractional Fokker-Planck-Smoluchowski equation display some very general and universal properties. These universal characteristics originate in the underlying competition between long jumps (fractional space derivative) and long waiting times (fractional time derivative). Using a few selected model examples the universal features of anomalous diffusion will be demonstrated. © 2013 Springer International Publishing Switzerland.

@article{
 title = {Non-Gaussian, non-dynamical stochastic resonance},
 type = {article},
 year = {2013},
 volume = {86},
 id = {0536f1c8-9063-36b9-a7bc-640d8a64f5ff},
 created = {2020-10-30T10:12:15.596Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.596Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {The classical model revealing stochastic resonance is a motion of an overdamped particle in a double-well fourth order potential when combined action of noise and external periodic driving results in amplifying of weak signals. Resonance behavior can also be observed in non-dynamical systems. The simplest example is a threshold triggered device. It consists of a periodic modulated input and noise. Every time an output crosses the threshold the signal is recorded. Such a digitally filtered signal is sensitive to the noise intensity. There exists the optimal value of the noise intensity resulting in the "most" periodic output. Here, we explore properties of the non-dynamical stochastic resonance in non-equilibrium situations, i.e. when the Gaussian noise is replaced by an α-stable noise. We demonstrate that non-equilibrium α-stable noises, depending on noise parameters, can either weaken or enhance the non-dynamical stochastic resonance. © 2013 The Author(s).},
 bibtype = {article},
 author = {Szczepaniec, K. and Dybiec, B.},
 doi = {10.1140/epjb/e2013-40619-8},
 journal = {European Physical Journal B},
 number = {11}
}

The classical model revealing stochastic resonance is a motion of an overdamped particle in a double-well fourth order potential when combined action of noise and external periodic driving results in amplifying of weak signals. Resonance behavior can also be observed in non-dynamical systems. The simplest example is a threshold triggered device. It consists of a periodic modulated input and noise. Every time an output crosses the threshold the signal is recorded. Such a digitally filtered signal is sensitive to the noise intensity. There exists the optimal value of the noise intensity resulting in the "most" periodic output. Here, we explore properties of the non-dynamical stochastic resonance in non-equilibrium situations, i.e. when the Gaussian noise is replaced by an α-stable noise. We demonstrate that non-equilibrium α-stable noises, depending on noise parameters, can either weaken or enhance the non-dynamical stochastic resonance. © 2013 The Author(s).

@article{
 title = {Axelrod model with extended conservativeness},
 type = {article},
 year = {2012},
 keywords = {Axelrod model,Social dynamics,extended conservativeness},
 volume = {23},
 id = {4c49c529-b307-3b4b-82e7-9d33d091257c},
 created = {2020-10-30T10:12:14.672Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:14.672Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {Similarity of opinions and memory about recent interactions are two main factors determining likelihood of social contacts. Here, we explore the Axelrod model with an extended conservativeness which incorporates not only similarity between individuals but also a preference to the last source of accepted information. The additional preference given to the last source of information increases the initial decay of the number of ideas in the system, changes the character of the phase transition between homogeneous and heterogeneous final states and could increase the number of stable regions (clusters) in the final state. © 2012 World Scientific Publishing Company.},
 bibtype = {article},
 author = {Dybiec, B.},
 doi = {10.1142/S0129183112500866},
 journal = {International Journal of Modern Physics C},
 number = {12}
}

Similarity of opinions and memory about recent interactions are two main factors determining likelihood of social contacts. Here, we explore the Axelrod model with an extended conservativeness which incorporates not only similarity between individuals but also a preference to the last source of accepted information. The additional preference given to the last source of information increases the initial decay of the number of ideas in the system, changes the character of the phase transition between homogeneous and heterogeneous final states and could increase the number of stable regions (clusters) in the final state. © 2012 World Scientific Publishing Company.

@article{
 title = {Axelrod model: Accepting or discussing},
 type = {article},
 year = {2012},
 volume = {85},
 id = {bfa677a9-a378-3d66-ba9e-31bec48560d9},
 created = {2020-10-30T10:12:16.463Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.463Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {Agents building social systems are characterized by complex states, and interactions among individuals can align their opinions. The Axelrod model describes how local interactions can result in emergence of cultural domains. We propose two variants of the Axelrod model where local consensus is reached either by listening and accepting one of neighbors' opinion or two agents discuss their opinion and achieve an agreement with mixed opinions. We show that the local agreement rule affects the character of the transition between the single culture and the multiculture regimes. © 2012 EDP Sciences, Società Italiana di Fisica, Springer-Verlag.},
 bibtype = {article},
 author = {Dybiec, B. and Mitarai, N. and Sneppen, K.},
 doi = {10.1140/epjb/e2012-30450-2},
 journal = {European Physical Journal B},
 number = {10}
}

Agents building social systems are characterized by complex states, and interactions among individuals can align their opinions. The Axelrod model describes how local interactions can result in emergence of cultural domains. We propose two variants of the Axelrod model where local consensus is reached either by listening and accepting one of neighbors' opinion or two agents discuss their opinion and achieve an agreement with mixed opinions. We show that the local agreement rule affects the character of the transition between the single culture and the multiculture regimes. © 2012 EDP Sciences, Società Italiana di Fisica, Springer-Verlag.

@article{
 title = {Fluctuation-dissipation relations under Lévy noises},
 type = {article},
 year = {2012},
 volume = {98},
 id = {ea3aab88-bc7c-3972-b49d-b92941c0442d},
 created = {2020-10-30T10:12:16.477Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.477Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {For systems close to equilibrium, the relaxation properties of measurable physical quantities are described by the linear response theory and the fluctuation-dissipation theorem (FDT). Accordingly, the response or the generalized susceptibility, which is a function of the unperturbed equilibrium system, can be related to the correlation between spontaneous fluctuations of a given conjugate variable. There have been several attempts to extend the FDT far from equilibrium, introducing new terms or using effective temperatures. Here, we discuss applicability of the generalized FDT to out-of-equilibrium systems perturbed by time-dependent deterministic forces and acting under the influence of white Lévy noise. For the linear and Gaussian case, the equilibrium correlation function provides a full description of the dynamic properties of the system. This is, however, no longer true for non-Gaussian Lévy noises, for which the second and sometimes also the first moments are divergent, indicating absence of underlying physical scales. This self-similar behavior of Lévy noises results in violation of the classical dissipation theorem for the stability index α<2. We show that by properly identifying appropriate variables conjugated to external perturbations and analyzing time-dependent distributions, the generalized FDT can be restored also for systems subject to Lévy noises. As a working example, we test the use of the generalized FDT for a linear system subject to Cauchy white noise. © 2012 Europhysics Letters Association.},
 bibtype = {article},
 author = {Dybiec, B. and Parrondo, J.M.R. and Gudowska-Nowak, E.},
 doi = {10.1209/0295-5075/98/50006},
 journal = {EPL},
 number = {5}
}

For systems close to equilibrium, the relaxation properties of measurable physical quantities are described by the linear response theory and the fluctuation-dissipation theorem (FDT). Accordingly, the response or the generalized susceptibility, which is a function of the unperturbed equilibrium system, can be related to the correlation between spontaneous fluctuations of a given conjugate variable. There have been several attempts to extend the FDT far from equilibrium, introducing new terms or using effective temperatures. Here, we discuss applicability of the generalized FDT to out-of-equilibrium systems perturbed by time-dependent deterministic forces and acting under the influence of white Lévy noise. For the linear and Gaussian case, the equilibrium correlation function provides a full description of the dynamic properties of the system. This is, however, no longer true for non-Gaussian Lévy noises, for which the second and sometimes also the first moments are divergent, indicating absence of underlying physical scales. This self-similar behavior of Lévy noises results in violation of the classical dissipation theorem for the stability index α<2. We show that by properly identifying appropriate variables conjugated to external perturbations and analyzing time-dependent distributions, the generalized FDT can be restored also for systems subject to Lévy noises. As a working example, we test the use of the generalized FDT for a linear system subject to Cauchy white noise. © 2012 Europhysics Letters Association.

@article{
 title = {Information spreading and development of cultural centers},
 type = {article},
 year = {2012},
 volume = {85},
 id = {173b80b4-5b09-33f8-9112-fcae3ced0887},
 created = {2020-10-30T10:12:16.541Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.541Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {The historical interplay between societies is governed by many factors, including in particular the spreading of languages, religion, and other symbolic traits. Cultural development, in turn, is coupled to the emergence and maintenance of information spreading. Strong centralized cultures exist due to attention from their members, whose faithfulness in turn relies on the supply of information. Here we discuss a culture evolution model on a planar geometry that takes into account aspects of the feedback between information spreading and its maintenance. Features of the model are highlighted by comparing it to cultural spreading in ancient and medieval Europe, where it suggests in particular that long-lived centers should be located in geographically remote regions. © 2012 American Physical Society.},
 bibtype = {article},
 author = {Dybiec, B. and Mitarai, N. and Sneppen, K.},
 doi = {10.1103/PhysRevE.85.056116},
 journal = {Physical Review E - Statistical, Nonlinear, and Soft Matter Physics},
 number = {5}
}

The historical interplay between societies is governed by many factors, including in particular the spreading of languages, religion, and other symbolic traits. Cultural development, in turn, is coupled to the emergence and maintenance of information spreading. Strong centralized cultures exist due to attention from their members, whose faithfulness in turn relies on the supply of information. Here we discuss a culture evolution model on a planar geometry that takes into account aspects of the feedback between information spreading and its maintenance. Features of the model are highlighted by comparing it to cultural spreading in ancient and medieval Europe, where it suggests in particular that long-lived centers should be located in geographically remote regions. © 2012 American Physical Society.

@book{
 title = {Stochastic diffusion and stable noise-induced phenomena},
 type = {book},
 year = {2011},
 source = {Fractional Dynamics: Recent Advances},
 id = {b620f931-4313-3530-9447-730ad4f88619},
 created = {2020-10-30T10:12:15.323Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.323Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2012 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. We discuss ubiquity of noise effects generated in non-equilibrium systems driven by “stable” random forces interpreted as limit cases of pure jump stochastic processes. In particular, such an approach leads to a generalization of the common Brownian motion by a L’evy diffusion process. This chapter briefly analyzes common properties of the anomalous transport and investigates the asymptotic relations between properly scaled continuous time random walks (CTRW) and fractional Smoluchowski–Fokker–Planck equations.},
 bibtype = {book},
 author = {Dybiec, B. and Gudowska-Nowak, E.},
 doi = {10.1142/9789814340595_0002}
}

© 2012 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. We discuss ubiquity of noise effects generated in non-equilibrium systems driven by “stable” random forces interpreted as limit cases of pure jump stochastic processes. In particular, such an approach leads to a generalization of the common Brownian motion by a L’evy diffusion process. This chapter briefly analyzes common properties of the anomalous transport and investigates the asymptotic relations between properly scaled continuous time random walks (CTRW) and fractional Smoluchowski–Fokker–Planck equations.

@article{
 title = {Harmonic oscillator under Lévy noise: Unexpected properties in the phase space},
 type = {article},
 year = {2011},
 volume = {83},
 id = {c1cd9523-4d32-3653-82fd-5b41f6e4779a},
 created = {2020-10-30T10:12:16.543Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.543Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {A harmonic oscillator under the influence of noise is a basic model of various physical phenomena. Under Gaussian white noise the position and velocity of the oscillator are independent random variables which are distributed according to the bivariate Gaussian distribution with elliptic level lines. The distribution of phase is homogeneous. None of these properties hold in the general Lévy case. Thus, the level lines of the joint probability density are not elliptic. The coordinate and the velocity of the oscillator are strongly dependent, and this dependence is quantified by introducing the corresponding parameter ("width deficit"). The distribution of the phase is inhomogeneous and highly nontrivial. © 2011 American Physical Society.},
 bibtype = {article},
 author = {Sokolov, I.M. and Ebeling, W. and Dybiec, B.},
 doi = {10.1103/PhysRevE.83.041118},
 journal = {Physical Review E - Statistical, Nonlinear, and Soft Matter Physics},
 number = {4}
}

A harmonic oscillator under the influence of noise is a basic model of various physical phenomena. Under Gaussian white noise the position and velocity of the oscillator are independent random variables which are distributed according to the bivariate Gaussian distribution with elliptic level lines. The distribution of phase is homogeneous. None of these properties hold in the general Lévy case. Thus, the level lines of the joint probability density are not elliptic. The coordinate and the velocity of the oscillator are strongly dependent, and this dependence is quantified by introducing the corresponding parameter ("width deficit"). The distribution of the phase is inhomogeneous and highly nontrivial. © 2011 American Physical Society.

@article{
 title = {Relaxation to stationary states for anomalous diffusion},
 type = {article},
 year = {2011},
 keywords = {Anomalous diffusion,Continuous time random walks,Fractional Fokker-Planck-Smoluchowski equation,Stochastic representation,Subordination},
 volume = {16},
 id = {ececf95c-e43b-38ac-8ef9-a3adca909c94},
 created = {2020-10-30T10:12:16.598Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.598Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {The fractional Fokker-Planck-Smoluchowski equation serves as a standard description of the anomalous diffusion. Within a current presentation we study properties of stationary states of the fractional Fokker-Planck-Smoluchowski equation in bounding potentials with special attention to the way in which stationary states are approached. It is demonstrated that the shape of the stationary state depends on exponents characterizing the jump length distributions and the external potential. The convergence rate to the stationary state can be of the double power-law type and is determined solely by the subdiffusion parameter. © 2011 Elsevier B.V.},
 bibtype = {article},
 author = {Dybiec, B. and Sokolov, I.M. and Chechkin, A.V.},
 doi = {10.1016/j.cnsns.2011.05.011},
 journal = {Communications in Nonlinear Science and Numerical Simulation},
 number = {12}
}

The fractional Fokker-Planck-Smoluchowski equation serves as a standard description of the anomalous diffusion. Within a current presentation we study properties of stationary states of the fractional Fokker-Planck-Smoluchowski equation in bounding potentials with special attention to the way in which stationary states are approached. It is demonstrated that the shape of the stationary state depends on exponents characterizing the jump length distributions and the external potential. The convergence rate to the stationary state can be of the double power-law type and is determined solely by the subdiffusion parameter. © 2011 Elsevier B.V.

@article{
 title = {Lévy ratchet in a weak noise limit: Theory and simulation},
 type = {article},
 year = {2011},
 volume = {191},
 id = {15545ad8-d820-317e-b744-9dd563c0cb32},
 created = {2020-10-30T10:12:17.318Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:17.318Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {We study the motion of a particle in a time-independent periodic potential with broken mirror symmetry under action of a Lévy-stable noise (Lévy ratchet). We develop an analytical approach to the problem based on the asymptotic probabilistic method of decomposition proposed by P. Imkeller and I. Pavlyukevich [J. Phys. A 39, L237 (2006); Stoch. Proc. Appl. 116, 611 (2006)]. We derive analytical expressions for the quantities characterizing the particle's motion, namely for the splitting probabilities of the first escape from a single well, for the transition probabilities to other wells and for the probability current. We pay particular attention to the interplay between the asymmetry of the ratchet potential and the asymmetry (skewness) of the Lévy noise. Extensive numerical simulations demonstrate a good agreement with the analytical predictions for sufficiently small intensities of the Lévy noise driving the particle. © 2011 EDP Sciences and Springer.},
 bibtype = {article},
 author = {Pavlyukevich, I. and Dybiec, B. and Chechkin, A.V. and Sokolov, I.M.},
 doi = {10.1140/epjst/e2010-01352-6},
 journal = {European Physical Journal: Special Topics},
 number = {1}
}

We study the motion of a particle in a time-independent periodic potential with broken mirror symmetry under action of a Lévy-stable noise (Lévy ratchet). We develop an analytical approach to the problem based on the asymptotic probabilistic method of decomposition proposed by P. Imkeller and I. Pavlyukevich [J. Phys. A 39, L237 (2006); Stoch. Proc. Appl. 116, 611 (2006)]. We derive analytical expressions for the quantities characterizing the particle's motion, namely for the splitting probabilities of the first escape from a single well, for the transition probabilities to other wells and for the probability current. We pay particular attention to the interplay between the asymmetry of the ratchet potential and the asymmetry (skewness) of the Lévy noise. Extensive numerical simulations demonstrate a good agreement with the analytical predictions for sufficiently small intensities of the Lévy noise driving the particle. © 2011 EDP Sciences and Springer.

@article{
 title = {Suppressing anomalous diffusion by cooperation},
 type = {article},
 year = {2010},
 volume = {43},
 id = {fbade175-f82c-351a-989d-5dec44f902f2},
 created = {2020-10-30T10:12:14.547Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:14.547Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {© 2010 IOP Publishing Ltd. Within a continuous time random walk scenario we consider a motion of a complex of particles which moves coherently. The motion of every particle is characterized by the waiting time and jump length distributions which are of the power-law type. Due to the interactions between particles it is assumed that the waiting time is adjusted to the shortest or to the longest waiting time. Analogously, the jump length is adjusted to the shortest or to the longest jump length. We show that adjustment to the shortest waiting time can suppress the subdiffusive behavior even in situations when the exponent characterizing the waiting time distribution assures subdiffusive motion of a single particle. Finally, we demonstrate that the characteristic of the motion depends on the number of particles building a complex.},
 bibtype = {article},
 author = {Dybiec, B.},
 doi = {10.1088/1751-8113/43/31/312001},
 journal = {Journal of Physics A: Mathematical and Theoretical},
 number = {31}
}

© 2010 IOP Publishing Ltd. Within a continuous time random walk scenario we consider a motion of a complex of particles which moves coherently. The motion of every particle is characterized by the waiting time and jump length distributions which are of the power-law type. Due to the interactions between particles it is assumed that the waiting time is adjusted to the shortest or to the longest waiting time. Analogously, the jump length is adjusted to the shortest or to the longest jump length. We show that adjustment to the shortest waiting time can suppress the subdiffusive behavior even in situations when the exponent characterizing the waiting time distribution assures subdiffusive motion of a single particle. Finally, we demonstrate that the characteristic of the motion depends on the number of particles building a complex.

@article{
 title = {Escape from the potential well: Competition between long jumps and long waiting times},
 type = {article},
 year = {2010},
 volume = {133},
 id = {229c2569-f51e-3946-818c-b1adfd806d89},
 created = {2020-10-30T10:12:14.672Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:14.672Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {Within a concept of the fractional diffusion equation and subordination, the paper examines the influence of a competition between long waiting times and long jumps on the escape from the potential well. Applying analytical arguments and numerical methods, we demonstrate that the presence of long waiting times distributed according to a power-law distribution with a diverging mean leads to very general asymptotic properties of the survival probability. The observed survival probability asymptotically decays like a power law whose form is not affected by the value of the exponent characterizing the power law jump length distribution. It is demonstrated that this behavior is typical of and generic for systems exhibiting long waiting times. We also show that the survival probability has a universal character not only asymptotically, but also at small times. Finally, it is indicated which properties of the first passage time density are sensitive to the exact value of the exponent characterizing the jump length distribution. © 2010 American Institute of Physics.},
 bibtype = {article},
 author = {Dybiec, B.},
 doi = {10.1063/1.3511722},
 journal = {Journal of Chemical Physics},
 number = {24}
}

Within a concept of the fractional diffusion equation and subordination, the paper examines the influence of a competition between long waiting times and long jumps on the escape from the potential well. Applying analytical arguments and numerical methods, we demonstrate that the presence of long waiting times distributed according to a power-law distribution with a diverging mean leads to very general asymptotic properties of the survival probability. The observed survival probability asymptotically decays like a power law whose form is not affected by the value of the exponent characterizing the power law jump length distribution. It is demonstrated that this behavior is typical of and generic for systems exhibiting long waiting times. We also show that the survival probability has a universal character not only asymptotically, but also at small times. Finally, it is indicated which properties of the first passage time density are sensitive to the exact value of the exponent characterizing the jump length distribution. © 2010 American Institute of Physics.

@article{
 title = {Universal character of escape kinetics from finite intervals},
 type = {article},
 year = {2010},
 volume = {41},
 id = {7e83decf-e649-3612-9686-ad9f9c2291f5},
 created = {2020-10-30T10:12:14.732Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:14.732Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {We study a motion of an anomalous random walker on finite intervals restricted by two absorbing boundaries. The competition between anomalously long jumps and long waiting times leads to a very general kind of behavior. Trapping events distributed according to the power-law distribution result in occurrence of the Mittag-Leffler decay pattern which in turn is responsible for universal asymptotic properties of escape kinetics. The presence of long jumps which can be distributed according to nonsymmetric heavy tailed distributions does not affect asymptotic properties of the survival probability. Therefore, the probability of finding a random walker within a domain of motion decays asymptotically according to the universal pattern derived from the Mittag-Leffler function, which describes decay of single modes in subdiffusive dynamics.},
 bibtype = {article},
 author = {Dybiec, B.},
 journal = {Acta Physica Polonica B},
 number = {5}
}

We study a motion of an anomalous random walker on finite intervals restricted by two absorbing boundaries. The competition between anomalously long jumps and long waiting times leads to a very general kind of behavior. Trapping events distributed according to the power-law distribution result in occurrence of the Mittag-Leffler decay pattern which in turn is responsible for universal asymptotic properties of escape kinetics. The presence of long jumps which can be distributed according to nonsymmetric heavy tailed distributions does not affect asymptotic properties of the survival probability. Therefore, the probability of finding a random walker within a domain of motion decays asymptotically according to the universal pattern derived from the Mittag-Leffler function, which describes decay of single modes in subdiffusive dynamics.

@article{
 title = {Approaching stationarity: Competition between long jumps and long waiting times},
 type = {article},
 year = {2010},
 keywords = {Diffusion,Stochastic particle dynamics (theory),Stochastic processes (experiment),Stochastic processes (theory)},
 volume = {2010},
 id = {454a1db7-5514-3588-a3e8-3156078aab07},
 created = {2020-10-30T10:12:14.738Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:14.738Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {Within the continuous-time random walk (CTRW) scenarios, properties of the overall motion are determined by the waiting time and the jump length distributions. In the decoupled case, with power-law distributed waiting times and jump lengths, the CTRW scenario is asymptotically described by the double (space and time) fractional Fokker-Planck equation. Properties of a system described by such an equation are determined by the subdiffusion parameter and the jump length exponent. Nevertheless, the stationary state is determined solely by the jump length distribution and the potential. The waiting time distribution determines only the rate of convergence to the stationary state. Here, we inspect the competition between long waiting times and long jumps and how this competition is reflected in the way in which a stationary state is reached. In particular, we show that the distance between a time-dependent and a stationary solution changes in time as a double power law. © 2010 IOP Publishing Ltd and SISSA.},
 bibtype = {article},
 author = {Dybiec, B.},
 doi = {10.1088/1742-5468/2010/03/P03019},
 journal = {Journal of Statistical Mechanics: Theory and Experiment},
 number = {3}
}

Within the continuous-time random walk (CTRW) scenarios, properties of the overall motion are determined by the waiting time and the jump length distributions. In the decoupled case, with power-law distributed waiting times and jump lengths, the CTRW scenario is asymptotically described by the double (space and time) fractional Fokker-Planck equation. Properties of a system described by such an equation are determined by the subdiffusion parameter and the jump length exponent. Nevertheless, the stationary state is determined solely by the jump length distribution and the potential. The waiting time distribution determines only the rate of convergence to the stationary state. Here, we inspect the competition between long waiting times and long jumps and how this competition is reflected in the way in which a stationary state is reached. In particular, we show that the distance between a time-dependent and a stationary solution changes in time as a double power law. © 2010 IOP Publishing Ltd and SISSA.

@article{
 title = {Anomalous diffusion on finite intervals},
 type = {article},
 year = {2010},
 keywords = {Stochastic particle dynamics (theory),Stochastic processes (experiment),Stochastic processes (theory),Transport properties (theory)},
 volume = {2010},
 id = {99860e19-d526-3d7a-a9bc-e662c3c1ea60},
 created = {2020-10-30T10:12:14.788Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:14.788Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {We study the properties of anomalous diffusion on finite intervals. The process studied due to the presence of trapping events and long jumps is described by a double-fractional (time and space) Fokker-Planck equation. The properties of the overall process are affected not only by long waiting times and long jumps but also by boundaries. Special attention is given to the examination of the survival probability and the first-passage-time density. Using analytical arguments and numerical methods, we show that the asymptotic form of the survival probability is determined by the trapping process. For a special choice of parameters, we compare numerical results with theoretical formulae, demonstrating that numerical solutions constructed by subordination methods reconstruct known analytical results very well. Finally, we show that the power-law distribution of waiting times is responsible for the divergence of the mean first-passage time even for a power-law distribution of jump lengths. © 2010 IOP Publishing Ltd.},
 bibtype = {article},
 author = {Dybiec, B.},
 doi = {10.1088/1742-5468/2010/01/P01011},
 journal = {Journal of Statistical Mechanics: Theory and Experiment},
 number = {1}
}

We study the properties of anomalous diffusion on finite intervals. The process studied due to the presence of trapping events and long jumps is described by a double-fractional (time and space) Fokker-Planck equation. The properties of the overall process are affected not only by long waiting times and long jumps but also by boundaries. Special attention is given to the examination of the survival probability and the first-passage-time density. Using analytical arguments and numerical methods, we show that the asymptotic form of the survival probability is determined by the trapping process. For a special choice of parameters, we compare numerical results with theoretical formulae, demonstrating that numerical solutions constructed by subordination methods reconstruct known analytical results very well. Finally, we show that the power-law distribution of waiting times is responsible for the divergence of the mean first-passage time even for a power-law distribution of jump lengths. © 2010 IOP Publishing Ltd.

@article{
 title = {Subordinated diffusion and continuous time random walk asymptotics},
 type = {article},
 year = {2010},
 volume = {20},
 id = {47927ebb-d2f7-3fad-97e7-8e36fd13fecb},
 created = {2020-10-30T10:12:15.636Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.636Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {Anomalous transport is usually described either by models of continuous time random walks (CTRWs) or, otherwise, by fractional Fokker-Planck equations (FFPEs). The asymptotic relation between properly scaled CTRW and fractional diffusion process has been worked out via various approaches widely discussed in literature. Here, we focus on a correspondence between CTRWs and time and space fractional diffusion equation stemming from two different methods aimed to accurately approximate anomalous diffusion processes. One of them is the Monte Carlo simulation of uncoupled CTRW with a Lévy α-stable distribution of jumps in space and a one-parameter Mittag-Leffler distribution of waiting times. The other is based on a discretized form of a subordinated Langevin equation in which the physical time defined via the number of subsequent steps of motion is itself a random variable. Both approaches are tested for their numerical performance and verified with known analytical solutions for the Green function of a space-time fractional diffusion equation. The comparison demonstrates a trade off between precision of constructed solutions and computational costs. The method based on the subordinated Langevin equation leads to a higher accuracy of results, while the CTRW framework with a Mittag-Leffler distribution of waiting times provides efficiently an approximate fundamental solution to the FFPE and converges to the probability density function of the subordinated process in a long-time limit. © 2010 American Institute of Physics.},
 bibtype = {article},
 author = {Dybiec, B. and Gudowska-Nowak, E.},
 doi = {10.1063/1.3522761},
 journal = {Chaos},
 number = {4}
}

Anomalous transport is usually described either by models of continuous time random walks (CTRWs) or, otherwise, by fractional Fokker-Planck equations (FFPEs). The asymptotic relation between properly scaled CTRW and fractional diffusion process has been worked out via various approaches widely discussed in literature. Here, we focus on a correspondence between CTRWs and time and space fractional diffusion equation stemming from two different methods aimed to accurately approximate anomalous diffusion processes. One of them is the Monte Carlo simulation of uncoupled CTRW with a Lévy α-stable distribution of jumps in space and a one-parameter Mittag-Leffler distribution of waiting times. The other is based on a discretized form of a subordinated Langevin equation in which the physical time defined via the number of subsequent steps of motion is itself a random variable. Both approaches are tested for their numerical performance and verified with known analytical solutions for the Green function of a space-time fractional diffusion equation. The comparison demonstrates a trade off between precision of constructed solutions and computational costs. The method based on the subordinated Langevin equation leads to a higher accuracy of results, while the CTRW framework with a Mittag-Leffler distribution of waiting times provides efficiently an approximate fundamental solution to the FFPE and converges to the probability density function of the subordinated process in a long-time limit. © 2010 American Institute of Physics.

@article{
 title = {Stationary states in single-well potentials under symmetric Lévy noises},
 type = {article},
 year = {2010},
 keywords = {Stationary states,Stochastic particle dynamics (theory),Stochastic processes (theory)},
 volume = {2010},
 id = {2e640f20-b9b3-31ce-b030-2e660e73ae3d},
 created = {2020-10-30T10:12:16.602Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.602Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {We discuss the existence of stationary states for subharmonic potentials V(x) ∝ |x|c, c < 2, under the action of symmetric α-stable noises. We show analytically that the necessary condition for the existence of the steady state is c > 2 - α. Consequently, for harmonic (c = 2) and superharmonic potentials (c > 2) driven by any α-stable noise, steady states always exist. Stationary states are characterized by probability density functions P(x) ∝ x-(c+α-1) for |x| → ∞ having a lighter tail than the noise distribution for superharmonic potentials (c > 2) and a heavier tail than the noise distribution for subharmonic ones. Monte Carlo simulations confirm the existence of such stationary states and the form of the tails of the corresponding probability densities. © 2010 IOP Publishing Ltd and SISSA.},
 bibtype = {article},
 author = {Dybiec, B. and Sokolov, I.M. and Chechkin, A.V.},
 doi = {10.1088/1742-5468/2010/07/P07008},
 journal = {Journal of Statistical Mechanics: Theory and Experiment},
 number = {7}
}

We discuss the existence of stationary states for subharmonic potentials V(x) ∝ |x|c, c < 2, under the action of symmetric α-stable noises. We show analytically that the necessary condition for the existence of the steady state is c > 2 - α. Consequently, for harmonic (c = 2) and superharmonic potentials (c > 2) driven by any α-stable noise, steady states always exist. Stationary states are characterized by probability density functions P(x) ∝ x-(c+α-1) for |x| → ∞ having a lighter tail than the noise distribution for superharmonic potentials (c > 2) and a heavier tail than the noise distribution for subharmonic ones. Monte Carlo simulations confirm the existence of such stationary states and the form of the tails of the corresponding probability densities. © 2010 IOP Publishing Ltd and SISSA.

@article{
 title = {Epidemics with short and long-range interactions: Role of vector dispersal patterns},
 type = {article},
 year = {2009},
 volume = {72},
 id = {85d82cee-91e2-3178-9fb9-bb42d7bb974c},
 created = {2020-10-30T10:12:14.827Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:14.827Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {We study the properties of the SIDRV (Susceptible-Infectious-Detected- Recovered-Vaccinated) model of the epidemic spread with local and non-local infection spread. The local spread is introduced by the nearest neighbors' interactions while the non-local spread is produced by vectors performing a random walk onto the system topology. Within the model we focus on a study of vectors' properties and on the interplay between vectors' characteristics and their dispersal patterns. We search for a type of a random walk which maximizes the time for which vectors are in the infectious state and consequently contribute to the (non-local) infection spread for the longest time. We also search for a type of a random walk which leads to the highest severity of epidemics. On the basis of numerical simulations we can conclude that from the whole considered class of random walks some are favored over others. We also show a very different performance and sensitivity of efficiency measures related to time and epidemics severity. Finally, we assess the role of assumptions taken within the model and discuss its relevance in designing elimination strategies, showing crucial role of local control strategies. © 2009 EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg.},
 bibtype = {article},
 author = {Dybiec, B.},
 doi = {10.1140/epjb/e2009-00403-1},
 journal = {European Physical Journal B},
 number = {4}
}

We study the properties of the SIDRV (Susceptible-Infectious-Detected- Recovered-Vaccinated) model of the epidemic spread with local and non-local infection spread. The local spread is introduced by the nearest neighbors' interactions while the non-local spread is produced by vectors performing a random walk onto the system topology. Within the model we focus on a study of vectors' properties and on the interplay between vectors' characteristics and their dispersal patterns. We search for a type of a random walk which maximizes the time for which vectors are in the infectious state and consequently contribute to the (non-local) infection spread for the longest time. We also search for a type of a random walk which leads to the highest severity of epidemics. On the basis of numerical simulations we can conclude that from the whole considered class of random walks some are favored over others. We also show a very different performance and sensitivity of efficiency measures related to time and epidemics severity. Finally, we assess the role of assumptions taken within the model and discuss its relevance in designing elimination strategies, showing crucial role of local control strategies. © 2009 EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg.

@article{
 title = {Anomalous diffusion: Temporal non-Markovianity and weak ergodicity breaking},
 type = {article},
 year = {2009},
 keywords = {Ergodicity breaking (theory),Stochastic particle dynamics (theory),Stochastic processes (experiment),Stochastic processes (theory)},
 volume = {2009},
 id = {b5988ac7-15d4-3b7e-8838-a56ac45aaf75},
 created = {2020-10-30T10:12:14.856Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:14.856Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {Traditionally, the discrimination between a Markovian and a non-Markovian process is based on the definition. If the process is Markovian, its transition probability does not depend on the history of the process and it fulfills the Smoluchowski-Chapman-Kolmogorov equation. A practical verification of these two criteria is not always possible or fully conclusive. Therefore, we present an additional method which can be used to confirm the simplest version of Markovianity. This method is based on the properties of sums of independent random variables. We apply the presented method to prove the increment dependent character of an anomalous process combining long waiting times with long jumps. Such a process, despite being non-Markovian in nature, due to a competition between long waiting times and long jumps, can reveal 'normal' behavior. We also demonstrate that this anomalous process breaks the ergodicity in the weak sense. Finally, we apply the suggested method to some experimental time series proving their Markovian nature for small timescales. © 2009 IOP Publishing Ltd and SISSA.},
 bibtype = {article},
 author = {Dybiec, B.},
 doi = {10.1088/1742-5468/2009/08/P08025},
 journal = {Journal of Statistical Mechanics: Theory and Experiment},
 number = {8}
}

Traditionally, the discrimination between a Markovian and a non-Markovian process is based on the definition. If the process is Markovian, its transition probability does not depend on the history of the process and it fulfills the Smoluchowski-Chapman-Kolmogorov equation. A practical verification of these two criteria is not always possible or fully conclusive. Therefore, we present an additional method which can be used to confirm the simplest version of Markovianity. This method is based on the properties of sums of independent random variables. We apply the presented method to prove the increment dependent character of an anomalous process combining long waiting times with long jumps. Such a process, despite being non-Markovian in nature, due to a competition between long waiting times and long jumps, can reveal 'normal' behavior. We also demonstrate that this anomalous process breaks the ergodicity in the weak sense. Finally, we apply the suggested method to some experimental time series proving their Markovian nature for small timescales. © 2009 IOP Publishing Ltd and SISSA.

@article{
 title = {Lévy noises: Double stochastic resonance in a single-well potential},
 type = {article},
 year = {2009},
 volume = {80},
 id = {981bb288-dddd-3a1d-a251-6a4781c4a837},
 created = {2020-10-30T10:12:14.874Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:14.874Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {We study properties of a single-well fourth-order potential perturbed by a periodically modulated stable noise. Periodic modulation of the stable noise asymmetry results in an occurrence of the dynamical hysteresis which is the manifestation of the stochastic resonance in the system at hand. We show that the single-well potential with time modulated stable driving is a minimalistic setup, allowing the occurrence of the stochastic resonance (as measured by the hysteresis loop area). Finally, we demonstrate that the observed stochastic resonance is of the double type, i.e., the system efficiency measured by the hysteresis loop area depends in a nonmonotonous way both on the scale parameter (noise intensity) and on the stability exponent characterizing tails asymptotic of noise pulses. © 2009 The American Physical Society.},
 bibtype = {article},
 author = {Dybiec, B.},
 doi = {10.1103/PhysRevE.80.041111},
 journal = {Physical Review E - Statistical, Nonlinear, and Soft Matter Physics},
 number = {4}
}

We study properties of a single-well fourth-order potential perturbed by a periodically modulated stable noise. Periodic modulation of the stable noise asymmetry results in an occurrence of the dynamical hysteresis which is the manifestation of the stochastic resonance in the system at hand. We show that the single-well potential with time modulated stable driving is a minimalistic setup, allowing the occurrence of the stochastic resonance (as measured by the hysteresis loop area). Finally, we demonstrate that the observed stochastic resonance is of the double type, i.e., the system efficiency measured by the hysteresis loop area depends in a nonmonotonous way both on the scale parameter (noise intensity) and on the stability exponent characterizing tails asymptotic of noise pulses. © 2009 The American Physical Society.

@article{
 title = {SIR model of epidemic spread with accumulated exposure},
 type = {article},
 year = {2009},
 volume = {67},
 id = {6e0a59a2-beb4-3e72-aecb-8ca7d70fe97a},
 created = {2020-10-30T10:12:14.912Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:14.912Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {We study an extended and modified SIR model of epidemic spread in which susceptible agents during interactions with infectious neighbors are exposed to the disease and can consequently become infectious. The studied model is extended to include heterogeneity of interactions which is modelled assuming random character of the dose accumulated by susceptible agents in every interaction with infectious neighbors. When the accumulated exposure is larger than the individual's resistance, an agent becomes infectious and consequently introduces a new source of an epidemic which is capable of passing the disease further. We study statistical properties characterizing the course of an epidemic. The examination of the modified SIR model reveals a possible "resonant activation"-like behavior of the system in the duration of the epidemic outbreak and a possible bistable behavior of the model with accumulated exposure. Furthermore, the linear scaling of the duration of the epidemic with the system size for a wide range of the model parameters is recorded. © 2008 EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg.},
 bibtype = {article},
 author = {Dybiec, B.},
 doi = {10.1140/epjb/e2008-00435-y},
 journal = {European Physical Journal B},
 number = {3}
}

We study an extended and modified SIR model of epidemic spread in which susceptible agents during interactions with infectious neighbors are exposed to the disease and can consequently become infectious. The studied model is extended to include heterogeneity of interactions which is modelled assuming random character of the dose accumulated by susceptible agents in every interaction with infectious neighbors. When the accumulated exposure is larger than the individual's resistance, an agent becomes infectious and consequently introduces a new source of an epidemic which is capable of passing the disease further. We study statistical properties characterizing the course of an epidemic. The examination of the modified SIR model reveals a possible "resonant activation"-like behavior of the system in the duration of the epidemic outbreak and a possible bistable behavior of the model with accumulated exposure. Furthermore, the linear scaling of the duration of the epidemic with the system size for a wide range of the model parameters is recorded. © 2008 EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg.

@article{
 title = {Anomalous diffusion and generalized Sparre Andersen scaling},
 type = {article},
 year = {2009},
 volume = {88},
 id = {450112fc-ab67-3efb-8c92-b2026f1abd29},
 created = {2020-10-30T10:12:15.644Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.644Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {We are discussing the long-time scaling limit for the anomalous diffusion composed of the subordinated Lévy-Wiener process. The limiting anomalous diffusion is in general non-Markov, even in the regime, where ensemble averages of a mean-square displacement or quantiles representing the group spread of the distribution follow the scaling characteristic for an ordinary stochastic diffusion. To discriminate between a truly memory-less process and the non-Markov one, we are analyzing the deviation of the survival probability from the (standard) Sparre Andersen scaling. Copyright © 2009 EPLA.},
 bibtype = {article},
 author = {Dybiec, B. and Gudowska-Nowak, E.},
 doi = {10.1209/0295-5075/88/10003},
 journal = {EPL},
 number = {1}
}

We are discussing the long-time scaling limit for the anomalous diffusion composed of the subordinated Lévy-Wiener process. The limiting anomalous diffusion is in general non-Markov, even in the regime, where ensemble averages of a mean-square displacement or quantiles representing the group spread of the distribution follow the scaling characteristic for an ordinary stochastic diffusion. To discriminate between a truly memory-less process and the non-Markov one, we are analyzing the deviation of the survival probability from the (standard) Sparre Andersen scaling. Copyright © 2009 EPLA.

@article{
 title = {Lévy stable noise-induced transitions: Stochastic resonance, resonant activation and dynamic hysteresis},
 type = {article},
 year = {2009},
 keywords = {Driven diffusive systems (theory),Stochastic particle dynamics(theory),Stochastic processes (theory),Transport processes/heat transfer (theory)},
 volume = {2009},
 id = {a410cd40-4f39-3c6d-a936-4fbd562a652c},
 created = {2020-10-30T10:12:15.720Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.720Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {A standard approach to analysis of noise-induced effects in stochastic dynamics assumes a Gaussian character of the noise term describing interaction of the analyzed system with its complex surroundings. An additional assumption about the existence of timescale separation between the dynamics of the measured observable and the typical timescale of the noise allows external fluctuations to be modeled as temporally uncorrelated and therefore white. However, in many natural phenomena the assumptions concerning the above mentioned properties of 'Gaussianity' and 'whiteness' of the noise can be violated. In this context, in contrast to the spatiotemporal coupling characterizing general forms of non-Markovian or semi-Markovian Lévy walks, so called Lévy flights correspond to the class of Markov processes which can still be interpreted as white, but distributed according to a more general, infinitely divisible, stable and non-Gaussian law. Lévy noise-driven non-equilibrium systems are known to manifest interesting physical properties and have been addressed in various scenarios of physical transport exhibiting a superdiffusive behavior. Here we present a brief overview of our recent investigations aimed at understanding features of stochastic dynamics under the influence of Lévy white noise perturbations. We find that the archetypal phenomena of noise-induced ordering are robust and can be detected also in systems driven by memoryless, non-Gaussian, heavy-tailed fluctuations with infinite variance. © 2009 IOP Publishing Ltd.},
 bibtype = {article},
 author = {Dybiec, B. and Gudowska-Nowak, E.},
 doi = {10.1088/1742-5468/2009/05/P05004},
 journal = {Journal of Statistical Mechanics: Theory and Experiment},
 number = {5}
}

A standard approach to analysis of noise-induced effects in stochastic dynamics assumes a Gaussian character of the noise term describing interaction of the analyzed system with its complex surroundings. An additional assumption about the existence of timescale separation between the dynamics of the measured observable and the typical timescale of the noise allows external fluctuations to be modeled as temporally uncorrelated and therefore white. However, in many natural phenomena the assumptions concerning the above mentioned properties of 'Gaussianity' and 'whiteness' of the noise can be violated. In this context, in contrast to the spatiotemporal coupling characterizing general forms of non-Markovian or semi-Markovian Lévy walks, so called Lévy flights correspond to the class of Markov processes which can still be interpreted as white, but distributed according to a more general, infinitely divisible, stable and non-Gaussian law. Lévy noise-driven non-equilibrium systems are known to manifest interesting physical properties and have been addressed in various scenarios of physical transport exhibiting a superdiffusive behavior. Here we present a brief overview of our recent investigations aimed at understanding features of stochastic dynamics under the influence of Lévy white noise perturbations. We find that the archetypal phenomena of noise-induced ordering are robust and can be detected also in systems driven by memoryless, non-Gaussian, heavy-tailed fluctuations with infinite variance. © 2009 IOP Publishing Ltd.

@article{
 title = {Discriminating between normal and anomalous random walks},
 type = {article},
 year = {2009},
 volume = {80},
 id = {81f39b26-97f0-37d6-8a32-4f0c34edfd79},
 created = {2020-10-30T10:12:15.746Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.746Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {Commonly, normal diffusive behavior is characterized by a linear dependence of the second central moment on time, x2 (t) t, while anomalous behavior is expected to show a different time dependence, x2 (t) tδ with δ<1 for subdiffusive and δ>1 for superdiffusive motions. Here we explore in details the fact that this kind of qualification, if applied straightforwardly, may be misleading: there are anomalous transport motions revealing perfectly "normal" diffusive character (x2 (t) t) yet being non-Markov and non-Gaussian in nature. We use recently developed framework of Monte Carlo simulations which incorporates anomalous diffusion statistics in time and space and creates trajectories of such an extended random walk. For special choice of stability indices describing statistics of waiting times and jump lengths, the ensemble analysis of anomalous diffusion is shown to hide temporal memory effects which can be properly detected only by examination of formal criteria of Markovianity (fulfillment of the Chapman-Kolmogorov equation). © 2009 The American Physical Society.},
 bibtype = {article},
 author = {Dybiec, B. and Gudowska-Nowak, E.},
 doi = {10.1103/PhysRevE.80.061122},
 journal = {Physical Review E - Statistical, Nonlinear, and Soft Matter Physics},
 number = {6}
}

Commonly, normal diffusive behavior is characterized by a linear dependence of the second central moment on time, x2 (t) t, while anomalous behavior is expected to show a different time dependence, x2 (t) tδ with δ<1 for subdiffusive and δ>1 for superdiffusive motions. Here we explore in details the fact that this kind of qualification, if applied straightforwardly, may be misleading: there are anomalous transport motions revealing perfectly "normal" diffusive character (x2 (t) t) yet being non-Markov and non-Gaussian in nature. We use recently developed framework of Monte Carlo simulations which incorporates anomalous diffusion statistics in time and space and creates trajectories of such an extended random walk. For special choice of stability indices describing statistics of waiting times and jump lengths, the ensemble analysis of anomalous diffusion is shown to hide temporal memory effects which can be properly detected only by examination of formal criteria of Markovianity (fulfillment of the Chapman-Kolmogorov equation). © 2009 The American Physical Society.

@article{
 title = {Modelling control of epidemics spreading by long-range interactions},
 type = {article},
 year = {2009},
 keywords = {Disease spread,Dispersal patterns,Epidemiological control,Epidemiological modelling,Stochastic modelling},
 volume = {6},
 id = {50c8661f-45fa-30a1-b98e-2a4d1933424d},
 created = {2020-10-30T10:12:16.679Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.679Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {We have studied the spread of epidemics characterized by a mixture of local and non-local interactions. The infection spreads on a two-dimensional lattice with the fixed nearest neighbour connections. In addition, long-range dynamical links are formed by moving agents (vectors). Vectors perform random walks, with step length distributed according to a thick-tail distribution. Two distributions are considered in this paper, an a-stable distribution describing self-similar vector movement, yet characterized by an infinite variance and an exponential power characterized by a large but finite variance. Such long-range interactions are hard to track and make control of epidemics very difficult. We also allowed for cryptic infection, whereby an infected individual on the lattice can be infectious prior to showing any symptoms of infection or disease. To account for such cryptic spread, we considered a control strategy in which not only detected, i.e. symptomatic, individuals but also all individuals within a certain control neighbourhood are treated upon the detection of disease. We show that it is possible to eradicate the disease by using such purely local control measures, even in the presence of long-range jumps. In particular, we show that the success of local control and the choice of the optimal strategy depend in a non-trivial way on the dispersal patterns of the vectors. By characterizing these patterns using the stability index of the a-stable distribution to change the power-law behaviour or the exponent characterizing the decay of an exponential power distribution, we show that infection can be successfully contained using relatively small control neighbourhoods for two limiting cases for long-distance dispersal and for vectors that are much more limited in their dispersal range. © 2009 The Royal Society.},
 bibtype = {article},
 author = {Dybiec, B. and Kleczkowski, A. and Gilligan, C.A.},
 doi = {10.1098/rsif.2008.0468},
 journal = {Journal of the Royal Society Interface},
 number = {39}
}

We have studied the spread of epidemics characterized by a mixture of local and non-local interactions. The infection spreads on a two-dimensional lattice with the fixed nearest neighbour connections. In addition, long-range dynamical links are formed by moving agents (vectors). Vectors perform random walks, with step length distributed according to a thick-tail distribution. Two distributions are considered in this paper, an a-stable distribution describing self-similar vector movement, yet characterized by an infinite variance and an exponential power characterized by a large but finite variance. Such long-range interactions are hard to track and make control of epidemics very difficult. We also allowed for cryptic infection, whereby an infected individual on the lattice can be infectious prior to showing any symptoms of infection or disease. To account for such cryptic spread, we considered a control strategy in which not only detected, i.e. symptomatic, individuals but also all individuals within a certain control neighbourhood are treated upon the detection of disease. We show that it is possible to eradicate the disease by using such purely local control measures, even in the presence of long-range jumps. In particular, we show that the success of local control and the choice of the optimal strategy depend in a non-trivial way on the dispersal patterns of the vectors. By characterizing these patterns using the stability index of the a-stable distribution to change the power-law behaviour or the exponent characterizing the decay of an exponential power distribution, we show that infection can be successfully contained using relatively small control neighbourhoods for two limiting cases for long-distance dispersal and for vectors that are much more limited in their dispersal range. © 2009 The Royal Society.

@article{
 title = {Stochastic diffusion: From Markov to non-Markov modeling},
 type = {article},
 year = {2009},
 volume = {40},
 id = {522f8144-7aac-3424-afc3-29bb51c66b78},
 created = {2020-10-30T10:12:17.350Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:17.350Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {We briefly discuss omnipresence of stochastic modeling in physical science by recalling definitions of Markovian diffusion and generally, non- Markovian continuous time random walks (CTRW). If the motion of an idealized system can be described by a sum of independent displacements whose statistic over short time intervals has a well defined variance, the resulting random walk converges to a normal diffusion process. In turn, if formulation of such motion assumes the idea of distribution of waiting times between subsequent steps, the CTRW scenario emerges, which typically violates the Markovian property.},
 bibtype = {article},
 author = {Gudowska-Nowak, E. and Dybiec, B. and Góra, P.F. and Zygadlo, R.},
 journal = {Acta Physica Polonica B},
 number = {5}
}

We briefly discuss omnipresence of stochastic modeling in physical science by recalling definitions of Markovian diffusion and generally, non- Markovian continuous time random walks (CTRW). If the motion of an idealized system can be described by a sum of independent displacements whose statistic over short time intervals has a well defined variance, the resulting random walk converges to a normal diffusion process. In turn, if formulation of such motion assumes the idea of distribution of waiting times between subsequent steps, the CTRW scenario emerges, which typically violates the Markovian property.

@article{
 title = {Current inversion in the Lévy ratchet},
 type = {article},
 year = {2008},
 volume = {78},
 id = {18156e3b-262f-3111-9d1a-6399824b4f53},
 created = {2020-10-30T10:12:14.924Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:14.924Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {We study the motion of an overdamped test particle in a static periodic potential lacking spatial symmetry under the influence of periodically modulated α -stable (Lévy) type noise. Due to the nonthermal character of the driving noise, the particle exhibits a motion with a preferred direction. The additional periodic modulation of the noise asymmetry changes the behavior of the static "Lévy ratchet." For the fast rate of the noise asymmetry modulation, the Lévy ratchet behaves like the one driven by the symmetric α -stable noise. When the modulation period is larger, the nontrivial effects of the noise asymmetry on the behavior of the Lévy ratchet are visible. In particular, the current inversion is observed in the system at hand. The properties of the Lévy ratchet are studied by use of the robust measures of directionality, which are defined regardless of the type of the stochastic driving. © 2008 The American Physical Society.},
 bibtype = {article},
 author = {Dybiec, B.},
 doi = {10.1103/PhysRevE.78.061120},
 journal = {Physical Review E - Statistical, Nonlinear, and Soft Matter Physics},
 number = {6}
}

We study the motion of an overdamped test particle in a static periodic potential lacking spatial symmetry under the influence of periodically modulated α -stable (Lévy) type noise. Due to the nonthermal character of the driving noise, the particle exhibits a motion with a preferred direction. The additional periodic modulation of the noise asymmetry changes the behavior of the static "Lévy ratchet." For the fast rate of the noise asymmetry modulation, the Lévy ratchet behaves like the one driven by the symmetric α -stable noise. When the modulation period is larger, the nontrivial effects of the noise asymmetry on the behavior of the Lévy ratchet are visible. In particular, the current inversion is observed in the system at hand. The properties of the Lévy ratchet are studied by use of the robust measures of directionality, which are defined regardless of the type of the stochastic driving. © 2008 The American Physical Society.

@article{
 title = {Random strategies of contact tracking},
 type = {article},
 year = {2008},
 keywords = {Lévy flights,Optimal search strategies,Random walks,α-stable Lévy type random variables},
 volume = {387},
 id = {6cc3f841-fd1e-3174-8c78-8c6607a7730e},
 created = {2020-10-30T10:12:14.961Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:14.961Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {One of several critical issues in the development of optimal disease containment and eradication strategies is the knowledge of underlying contacts between individuals. Here we employ random search strategies to identify all possible links, representing direct or indirect interactions between individuals building up the system. In order to recognize all contacts, the searcher performs symmetric Lévy flights onto the accessible area. We investigate the influence of local and non-local information, the exponent characterizing asymptotic behavior of Lévy flights, boundary conditions, density of links and type of a search strategy on the efficiency of the search process. Monte Carlo examination of the suggested model reveals that the efficiency of the search process is sensitive to the type of boundary conditions. Depending on the assumed type of boundary conditions, efficiency of the search process can be a monotonic or non-monotonic function of the exponents characterizing asymptotic behavior of Lévy flights. Consequently, among the whole spectrum of exponents characterizing the power law behavior of jumps' length, there exist distinguished values of stability index representing the most efficient search processes. These exponents correspond to extreme (minimal or maximal) or intermediate values of stability index associated with Gaussian, maximally heavy-tailed or Cauchy-like strategies, respectively. © 2008 Elsevier B.V. All rights reserved.},
 bibtype = {article},
 author = {Dybiec, B.},
 doi = {10.1016/j.physa.2008.04.027},
 journal = {Physica A: Statistical Mechanics and its Applications},
 number = {19-20}
}

One of several critical issues in the development of optimal disease containment and eradication strategies is the knowledge of underlying contacts between individuals. Here we employ random search strategies to identify all possible links, representing direct or indirect interactions between individuals building up the system. In order to recognize all contacts, the searcher performs symmetric Lévy flights onto the accessible area. We investigate the influence of local and non-local information, the exponent characterizing asymptotic behavior of Lévy flights, boundary conditions, density of links and type of a search strategy on the efficiency of the search process. Monte Carlo examination of the suggested model reveals that the efficiency of the search process is sensitive to the type of boundary conditions. Depending on the assumed type of boundary conditions, efficiency of the search process can be a monotonic or non-monotonic function of the exponents characterizing asymptotic behavior of Lévy flights. Consequently, among the whole spectrum of exponents characterizing the power law behavior of jumps' length, there exist distinguished values of stability index representing the most efficient search processes. These exponents correspond to extreme (minimal or maximal) or intermediate values of stability index associated with Gaussian, maximally heavy-tailed or Cauchy-like strategies, respectively. © 2008 Elsevier B.V. All rights reserved.

@article{
 title = {Barrier crossing process driven by two dichotomous noises},
 type = {article},
 year = {2008},
 keywords = {Dichotomous noise,First-passage problems,Resonant activation,Stochastic dierential equations,Stochastic methods},
 volume = {19},
 id = {9e1be31b-b0e2-3072-a3e6-731d741e56dc},
 created = {2020-10-30T10:12:14.994Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:14.994Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {We study the motion of an overdamped Brownian particle in a linear potential between reflecting and absorbing boundaries. The moving particle experiences the combined action of random and deterministic forces. In the model, the well-known and celebrated phenomenon of resonant activation is visible. The presence of two dichotomous noises modulates the shape of the potential barrier and changes the force acting on the particle leading to further acceleration of the escape kinetics in comparison to the situation where only one dichotomous process is present. In the case of very small or very large correlation times of dichotomous noises, we can observe asymptotic behavior of the system in which effectively one of the dichotomous noises is eliminated. © 2008 World Scientific Publishing Company.},
 bibtype = {article},
 author = {Dybiec, B.},
 doi = {10.1142/S0129183108011930},
 journal = {International Journal of Modern Physics C},
 number = {1}
}

We study the motion of an overdamped Brownian particle in a linear potential between reflecting and absorbing boundaries. The moving particle experiences the combined action of random and deterministic forces. In the model, the well-known and celebrated phenomenon of resonant activation is visible. The presence of two dichotomous noises modulates the shape of the potential barrier and changes the force acting on the particle leading to further acceleration of the escape kinetics in comparison to the situation where only one dichotomous process is present. In the case of very small or very large correlation times of dichotomous noises, we can observe asymptotic behavior of the system in which effectively one of the dichotomous noises is eliminated. © 2008 World Scientific Publishing Company.

@article{
 title = {Transport in a Lévy ratchet: Group velocity and distribution spread},
 type = {article},
 year = {2008},
 volume = {78},
 id = {aad4f48c-80ff-3480-9bb6-2f9fa4cb0c05},
 created = {2020-10-30T10:12:16.761Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.761Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {We consider the motion of an overdamped particle in a periodic potential lacking spatial symmetry under the influence of symmetric, white, Lévy noise, being a minimal setup for a "Lévy ratchet." Due to the nonthermal character of the Lévy noise, the particle exhibits a motion with a preferred direction even in the absence of whatever additional time-dependent forces. The examination of the Lévy ratchet has to be based on the characteristics of directionality which are different from typically used measures such as mean current and the dispersion of particle positions, since these become inappropriate when the moments of the noise diverge. To overcome this problem, we discuss robust measures of directionality of transport such as the position of the median of the particle displacement distribution characterizing the group velocity and the interquantile distance giving the measure of the distribution width. Moreover, we analyze the behavior of splitting probabilities for leaving an interval of a given length, unveiling qualitative differences between the noises with Lévy indices below and above unity. © 2008 The American Physical Society.},
 bibtype = {article},
 author = {Dybiec, B. and Gudowska-Nowak, E. and Sokolov, I.M.},
 doi = {10.1103/PhysRevE.78.011117},
 journal = {Physical Review E - Statistical, Nonlinear, and Soft Matter Physics},
 number = {1}
}

We consider the motion of an overdamped particle in a periodic potential lacking spatial symmetry under the influence of symmetric, white, Lévy noise, being a minimal setup for a "Lévy ratchet." Due to the nonthermal character of the Lévy noise, the particle exhibits a motion with a preferred direction even in the absence of whatever additional time-dependent forces. The examination of the Lévy ratchet has to be based on the characteristics of directionality which are different from typically used measures such as mean current and the dispersion of particle positions, since these become inappropriate when the moments of the noise diverge. To overcome this problem, we discuss robust measures of directionality of transport such as the position of the median of the particle displacement distribution characterizing the group velocity and the interquantile distance giving the measure of the distribution width. Moreover, we analyze the behavior of splitting probabilities for leaving an interval of a given length, unveiling qualitative differences between the noises with Lévy indices below and above unity. © 2008 The American Physical Society.

@article{
 title = {Bimodality and hysteresis in systems driven by confined Lévy flights},
 type = {article},
 year = {2007},
 volume = {9},
 id = {edf76910-0dd3-3967-9c9b-6bfc6c7377ed},
 created = {2020-10-30T10:12:15.847Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.847Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {We demonstrate the occurrence of bimodality and dynamical hysteresis in a system describing an overdamped quartic oscillator perturbed by additive white and asymmetric Lévy noise. Investigated estimators of the stationary probability density profiles display not only a turnover from unimodal to bimodal character but also a change in a relative stability of stationary states that depends on the asymmetry parameter of the underlying noise term. When varying the asymmetry parameter cyclically, the system exhibits a hysteresis in the occupation of a chosen stationary state. © IOP Publishing Ltd and Deutsche Physikalische Gesellscnaft.},
 bibtype = {article},
 author = {Dybiec, B. and Gudowska-Nowak, E.},
 doi = {10.1088/1367-2630/9/12/452},
 journal = {New Journal of Physics}
}

We demonstrate the occurrence of bimodality and dynamical hysteresis in a system describing an overdamped quartic oscillator perturbed by additive white and asymmetric Lévy noise. Investigated estimators of the stationary probability density profiles display not only a turnover from unimodal to bimodal character but also a change in a relative stability of stationary states that depends on the asymmetry parameter of the underlying noise term. When varying the asymmetry parameter cyclically, the system exhibits a hysteresis in the occupation of a chosen stationary state. © IOP Publishing Ltd and Deutsche Physikalische Gesellscnaft.

@article{
 title = {Quantifying noise induced effects in the generic double-well potential},
 type = {article},
 year = {2007},
 volume = {38},
 id = {730e1482-58b9-38e3-95a0-dc3714e14d85},
 created = {2020-10-30T10:12:15.872Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.872Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {Contrary to conventional wisdom, the transmission and detection of signals, efficiency of kinetics in the presence of fluctuating barriers or system's synchronization to the applied driving may be enhanced by random noise. We have numerically analyzed effects of the addition of external noise to a dynamical system representing a bistable over-damped oscillator and detected constructive influence of noise in the phenomena of resonant activation (RA), stochastic resonance (SR), dynamical hysteresis and noiseinduced stability (NES), We have documented that all above-mentioned effects can be observed in the very same system, although for slightly different regimes of parameters characterizing external periodic driving or (and) noise. Particular emphasis has been given to presentation of various quantifiers of the noise-induced constructive phenomena and their sensitivity to the location and character of the imposed boundary condition.},
 bibtype = {article},
 author = {Dybiec, B. and Gudowska-Nowak, E.},
 journal = {Acta Physica Polonica B},
 number = {5}
}

Contrary to conventional wisdom, the transmission and detection of signals, efficiency of kinetics in the presence of fluctuating barriers or system's synchronization to the applied driving may be enhanced by random noise. We have numerically analyzed effects of the addition of external noise to a dynamical system representing a bistable over-damped oscillator and detected constructive influence of noise in the phenomena of resonant activation (RA), stochastic resonance (SR), dynamical hysteresis and noiseinduced stability (NES), We have documented that all above-mentioned effects can be observed in the very same system, although for slightly different regimes of parameters characterizing external periodic driving or (and) noise. Particular emphasis has been given to presentation of various quantifiers of the noise-induced constructive phenomena and their sensitivity to the location and character of the imposed boundary condition.

@article{
 title = {Emergence of bimodality in noisy systems with single-well potential},
 type = {article},
 year = {2007},
 volume = {57},
 id = {d0c181a7-47cb-33d7-a7d2-941fbbe6cddc},
 created = {2020-10-30T10:12:15.909Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.909Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {We study the stationary probability density of a Brownian particle in a potential with a single-well subject to the purely additive thermal and dichotomous noise sources. We find situations where bimodality of stationary densities emerges due to presence of dichotomous noise. The solutions are constructed using stochastic dynamics (Langevin equation) or by discretization of the corresponding Fokker-Planck equations. We find that in models with both noises being additive the potential has to grow faster than |x| in order to obtain bimodality. For potentials ∞|x| stationary solutions are always of the double exponential form. © EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007.},
 bibtype = {article},
 author = {Dybiec, B. and Schimansky-Geier, L.},
 doi = {10.1140/epjb/e2007-00162-y},
 journal = {European Physical Journal B},
 number = {3}
}

We study the stationary probability density of a Brownian particle in a potential with a single-well subject to the purely additive thermal and dichotomous noise sources. We find situations where bimodality of stationary densities emerges due to presence of dichotomous noise. The solutions are constructed using stochastic dynamics (Langevin equation) or by discretization of the corresponding Fokker-Planck equations. We find that in models with both noises being additive the potential has to grow faster than |x| in order to obtain bimodality. For potentials ∞|x| stationary solutions are always of the double exponential form. © EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007.

@article{
 title = {Stationary states in Langevin dynamics under asymmetric Lévy noises},
 type = {article},
 year = {2007},
 volume = {76},
 id = {08e9df8e-3ad1-333b-b1bd-e7cfe09ee5b7},
 created = {2020-10-30T10:12:16.855Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.855Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {Properties of systems driven by white non-Gaussian noises can be very different from these of systems driven by the white Gaussian noise. We investigate stationary probability densities for systems driven by α -stable Lévy-type noises, which provide natural extension to the Gaussian noise having, however, a new property, namely a possibility of being asymmetric. Stationary probability densities are examined for a particle moving in parabolic, quartic, and in generic double well potential models subjected to the action of α -stable noises. Relevant solutions are constructed by methods of stochastic dynamics. In situations where analytical results are known they are compared with numerical results. Furthermore, the problem of estimation of the parameters of stationary densities is investigated. © 2007 The American Physical Society.},
 bibtype = {article},
 author = {Dybiec, B. and Gudowska-Nowak, E. and Sokolov, I.M.},
 doi = {10.1103/PhysRevE.76.041122},
 journal = {Physical Review E - Statistical, Nonlinear, and Soft Matter Physics},
 number = {4}
}

Properties of systems driven by white non-Gaussian noises can be very different from these of systems driven by the white Gaussian noise. We investigate stationary probability densities for systems driven by α -stable Lévy-type noises, which provide natural extension to the Gaussian noise having, however, a new property, namely a possibility of being asymmetric. Stationary probability densities are examined for a particle moving in parabolic, quartic, and in generic double well potential models subjected to the action of α -stable noises. Relevant solutions are constructed by methods of stochastic dynamics. In situations where analytical results are known they are compared with numerical results. Furthermore, the problem of estimation of the parameters of stationary densities is investigated. © 2007 The American Physical Society.

@article{
 title = {Escape driven by α -stable white noises},
 type = {article},
 year = {2007},
 volume = {75},
 id = {196c55a0-5536-3c91-9dfd-ab2f79f8195e},
 created = {2020-10-30T10:12:16.876Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.876Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {We explore the archetype problem of an escape dynamics occurring in a symmetric double well potential when the Brownian particle is driven by white Lévy noise in a dynamical regime where inertial effects can safely be neglected. The behavior of escaping trajectories from one well to another is investigated by pointing to the special character that underpins the noise-induced discontinuity which is caused by the generalized Brownian paths that jump beyond the barrier location without actually hitting it. This fact implies that the boundary conditions for the mean first passage time (MFPT) are no longer determined by the well-known local boundary conditions that characterize the case with normal diffusion. By numerically implementing properly the set up boundary conditions, we investigate the survival probability and the average escape time as a function of the corresponding Lévy white noise parameters. Depending on the value of the skewness β of the Lévy noise, the escape can either become enhanced or suppressed: a negative asymmetry parameter β typically yields a decrease for the escape rate while the rate itself depicts a non-monotonic behavior as a function of the stability index α that characterizes the jump length distribution of Lévy noise, exhibiting a marked discontinuity at α=1. We find that the typical factor of 2 that characterizes for normal diffusion the ratio between the MFPT for well-bottom-to-well-bottom and well-bottom-to-barrier-top no longer holds true. For sufficiently high barriers the survival probabilities assume an exponential behavior versus time. Distinct non-exponential deviations occur, however, for low barrier heights. © 2007 The American Physical Society.},
 bibtype = {article},
 author = {Dybiec, B. and Gudowska-Nowak, E. and Hänggi, P.},
 doi = {10.1103/PhysRevE.75.021109},
 journal = {Physical Review E - Statistical, Nonlinear, and Soft Matter Physics},
 number = {2}
}

We explore the archetype problem of an escape dynamics occurring in a symmetric double well potential when the Brownian particle is driven by white Lévy noise in a dynamical regime where inertial effects can safely be neglected. The behavior of escaping trajectories from one well to another is investigated by pointing to the special character that underpins the noise-induced discontinuity which is caused by the generalized Brownian paths that jump beyond the barrier location without actually hitting it. This fact implies that the boundary conditions for the mean first passage time (MFPT) are no longer determined by the well-known local boundary conditions that characterize the case with normal diffusion. By numerically implementing properly the set up boundary conditions, we investigate the survival probability and the average escape time as a function of the corresponding Lévy white noise parameters. Depending on the value of the skewness β of the Lévy noise, the escape can either become enhanced or suppressed: a negative asymmetry parameter β typically yields a decrease for the escape rate while the rate itself depicts a non-monotonic behavior as a function of the stability index α that characterizes the jump length distribution of Lévy noise, exhibiting a marked discontinuity at α=1. We find that the typical factor of 2 that characterizes for normal diffusion the ratio between the MFPT for well-bottom-to-well-bottom and well-bottom-to-barrier-top no longer holds true. For sufficiently high barriers the survival probabilities assume an exponential behavior versus time. Distinct non-exponential deviations occur, however, for low barrier heights. © 2007 The American Physical Society.

@inproceedings{
 title = {Stochastic resonance: The role of α-stable noises},
 type = {inproceedings},
 year = {2006},
 volume = {37},
 issue = {5},
 id = {a9a54ef5-e529-3daf-bc1c-d0563f21a4b2},
 created = {2020-10-30T10:12:15.940Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.940Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {In order to document and discuss the widespread presence in nature of the stochastic resonance phenomenon (SR), we investigate the generic double-well potential model perturbed by the α-stable Lévy type noises. Despite possible infinite variance characteristics of the noise term, the SR effect still occurs and can be detected by common quantifiers used in the studies of the phenomenon. The robustness of the SR is examined by use of standard measures within a continuous and a two-state description of the system. Since the α-stable noises are characterized by a whole set of parameters, our research focuses on the analysis of the influence of noise parameters on a shape of the signal-to-noise ratio and spectral power amplification curves, revealing presence of the SR in the system at hand. Examination of the driving noise asymmetry indicates that the resonant response weakens when the symmetric noises with a decreasing value of the stability index α are applied.},
 bibtype = {inproceedings},
 author = {Dybiec, B. and Gudowska-Nowak, E.},
 booktitle = {Acta Physica Polonica B}
}

In order to document and discuss the widespread presence in nature of the stochastic resonance phenomenon (SR), we investigate the generic double-well potential model perturbed by the α-stable Lévy type noises. Despite possible infinite variance characteristics of the noise term, the SR effect still occurs and can be detected by common quantifiers used in the studies of the phenomenon. The robustness of the SR is examined by use of standard measures within a continuous and a two-state description of the system. Since the α-stable noises are characterized by a whole set of parameters, our research focuses on the analysis of the influence of noise parameters on a shape of the signal-to-noise ratio and spectral power amplification curves, revealing presence of the SR in the system at hand. Examination of the driving noise asymmetry indicates that the resonant response weakens when the symmetric noises with a decreasing value of the stability index α are applied.

@inproceedings{
 title = {Economic and social factors in designing disease control strategies for epidemics on networks},
 type = {inproceedings},
 year = {2006},
 volume = {37},
 issue = {11},
 id = {649d57be-973e-3707-9c9e-76db83568d9f},
 created = {2020-10-30T10:12:16.910Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.910Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {Models for control of epidemics on local, global and small-world networks are considered, with only partial information accessible about the status of individuals and their connections. The main goal of an effective control measure is to stop the epidemic at a lowest possible cost, including treatment and cost necessary to track the disease spread. We show that delay in detection of infectious individuals and presence of long-range links are the most important factors determining the cost. However, the details of long-range links are usually the least-known element of the social interactions due to their occasional character and potentially short life-span. We show that under some conditions on the probability of disease spread, it is advisable to attempt to track those links, even if this involves additional costs. Thus, collecting some additional knowledge about the network structure might be beneficial to ensure a successful and cost-effective control.},
 bibtype = {inproceedings},
 author = {Kleczkowski, A. and Dybiec, B. and Gilligan, C.A.},
 booktitle = {Acta Physica Polonica B}
}

Models for control of epidemics on local, global and small-world networks are considered, with only partial information accessible about the status of individuals and their connections. The main goal of an effective control measure is to stop the epidemic at a lowest possible cost, including treatment and cost necessary to track the disease spread. We show that delay in detection of infectious individuals and presence of long-range links are the most important factors determining the cost. However, the details of long-range links are usually the least-known element of the social interactions due to their occasional character and potentially short life-span. We show that under some conditions on the probability of disease spread, it is advisable to attempt to track those links, even if this involves additional costs. Thus, collecting some additional knowledge about the network structure might be beneficial to ensure a successful and cost-effective control.

@article{
 title = {Lévy-Brownian motion on finite intervals: Mean first passage time analysis},
 type = {article},
 year = {2006},
 volume = {73},
 id = {22e77857-5982-3137-b313-47862e0d53d2},
 created = {2020-10-30T10:12:16.933Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.933Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {We present the analysis of the first passage time problem on a finite interval for the generalized Wiener process that is driven by Lévy stable noises. The complexity of the first passage time statistics (mean first passage time, cumulative first passage time distribution) is elucidated together with a discussion of the proper setup of corresponding boundary conditions that correctly yield the statistics of first passages for these non-Gaussian noises. The validity of the method is tested numerically and compared against analytical formulas when the stability index α approaches 2, recovering in this limit the standard results for the Fokker-Planck dynamics driven by Gaussian white noise. © 2006 The American Physical Society.},
 bibtype = {article},
 author = {Dybiec, B. and Gudowska-Nowak, E. and Hänggi, P.},
 doi = {10.1103/PhysRevE.73.046104},
 journal = {Physical Review E - Statistical, Nonlinear, and Soft Matter Physics},
 number = {4}
}

We present the analysis of the first passage time problem on a finite interval for the generalized Wiener process that is driven by Lévy stable noises. The complexity of the first passage time statistics (mean first passage time, cumulative first passage time distribution) is elucidated together with a discussion of the proper setup of corresponding boundary conditions that correctly yield the statistics of first passages for these non-Gaussian noises. The validity of the method is tested numerically and compared against analytical formulas when the stability index α approaches 2, recovering in this limit the standard results for the Fokker-Planck dynamics driven by Gaussian white noise. © 2006 The American Physical Society.

@article{
 title = {Stochastic resonance vs. resonant activation},
 type = {article},
 year = {2006},
 volume = {74},
 id = {9bbae7d2-cf1e-39cd-a782-4f4d26027f30},
 created = {2020-10-30T10:12:17.420Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:17.420Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {The phenomenon of stochastic resonance by which it is possible to boost the transduction of information by tuning into noise is experimentally and numerically contrasted with the phenomenon of resonant activation, i.e. the occurrence of a minimal, averaged residence time at an optimal time-scale in the presence of barrier modulations. The experimental system consists of a periodically modulated bistable colloidal Brownian dynamics. Interestingly enough, the two phenomena may occur simultaneously in the same system, although typically in quite different parameter regimes. © EDP Sciences.},
 bibtype = {article},
 author = {Schmitt, C. and Dybiec, B. and Hänggi, P. and Bechinger, C.},
 doi = {10.1209/epl/i2006-10052-6},
 journal = {Europhysics Letters},
 number = {6}
}

The phenomenon of stochastic resonance by which it is possible to boost the transduction of information by tuning into noise is experimentally and numerically contrasted with the phenomenon of resonant activation, i.e. the occurrence of a minimal, averaged residence time at an optimal time-scale in the presence of barrier modulations. The experimental system consists of a periodically modulated bistable colloidal Brownian dynamics. Interestingly enough, the two phenomena may occur simultaneously in the same system, although typically in quite different parameter regimes. © EDP Sciences.

@inproceedings{
 title = {Optimising control of disease spread on networks},
 type = {inproceedings},
 year = {2005},
 volume = {36},
 issue = {5},
 id = {0ee0db94-5964-3265-922d-29270b313e0c},
 created = {2020-10-30T10:12:16.962Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.962Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {We consider models for control of epidemics on local, global, small-world and scale-free networks, with only partial information accessible about the status of individuals and their connections. The effectiveness of local (e.g. ring vaccination or culling) vs global (e.g. random vaccination) control measures is evaluated, with the aim of minimising the total cost of an epidemic. The costs include direct costs of treating infected individuals as well as costs of treatment. We first consider a random (global) vaccination strategy designed to stop any potential outbreak. We show that if the costs of the preventive vaccination are ignored, the optimal strategy is to vaccinate the whole population, although most of the resources are wasted on preventing a small number of cases. If the vaccination costs are included, or if a local strategy (within a certain neighbourhood of a symptomatic individual) is chosen, there is an optimum number of treated individuals. Inclusion of non-local contacts ('small-worlds' or scale-free networks) increases the levels of preventive (random) vaccination and radius of local treatment necessary for stopping the outbreak at a minimal cost. The number of initial foci also influences our choice of optimal strategy. The size of epidemics and the number of treated individuals increase for outbreaks that are initiated from a larger number of initial foci, but the optimal radius of local control actually decreases. The results are important for designing control strategies based on cost effectiveness.},
 bibtype = {inproceedings},
 author = {Dybiec, B. and Kleczkowski, A. and Gilligan, C.A.},
 booktitle = {Acta Physica Polonica B}
}

We consider models for control of epidemics on local, global, small-world and scale-free networks, with only partial information accessible about the status of individuals and their connections. The effectiveness of local (e.g. ring vaccination or culling) vs global (e.g. random vaccination) control measures is evaluated, with the aim of minimising the total cost of an epidemic. The costs include direct costs of treating infected individuals as well as costs of treatment. We first consider a random (global) vaccination strategy designed to stop any potential outbreak. We show that if the costs of the preventive vaccination are ignored, the optimal strategy is to vaccinate the whole population, although most of the resources are wasted on preventing a small number of cases. If the vaccination costs are included, or if a local strategy (within a certain neighbourhood of a symptomatic individual) is chosen, there is an optimum number of treated individuals. Inclusion of non-local contacts ('small-worlds' or scale-free networks) increases the levels of preventive (random) vaccination and radius of local treatment necessary for stopping the outbreak at a minimal cost. The number of initial foci also influences our choice of optimal strategy. The size of epidemics and the number of treated individuals increase for outbreaks that are initiated from a larger number of initial foci, but the optimal radius of local control actually decreases. The results are important for designing control strategies based on cost effectiveness.

@article{
 title = {Hysteresis and synchronization in a two-level system driven by external noise},
 type = {article},
 year = {2005},
 keywords = {Mean first passage time,Resonance,Resonant activation,Stochastic hysteresis,Synchronization},
 volume = {5},
 id = {dc813689-af67-3dec-a57d-218c4030621f},
 created = {2020-10-30T10:12:17.000Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:17.000Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {The paper discusses a combined effect of periodic and random perturbations on the onset of hysteresis in a generic two-state system. The interplay of both noises is investigated pointing out variations in signal tuning to the external driving force and their influence on the area and shape of the hysteresis. As an extension to former studies relating dynamic hysteresis to stochastic resonance and synchronization of passages in a double well system, in the present work the effect of the field asymmetry on the response of the system is analyzed. © World Scientific Publishing Company.},
 bibtype = {article},
 author = {Juraszek, J. and Dybiec, B. and Gudowska-Nowak, E.},
 doi = {10.1142/S021947750500263X},
 journal = {Fluctuation and Noise Letters},
 number = {2}
}

The paper discusses a combined effect of periodic and random perturbations on the onset of hysteresis in a generic two-state system. The interplay of both noises is investigated pointing out variations in signal tuning to the external driving force and their influence on the area and shape of the hysteresis. As an extension to former studies relating dynamic hysteresis to stochastic resonance and synchronization of passages in a double well system, in the present work the effect of the field asymmetry on the response of the system is analyzed. © World Scientific Publishing Company.

@article{
 title = {Resonant activation in the presence of nonequilibrated baths},
 type = {article},
 year = {2004},
 volume = {69},
 id = {a9708033-5390-3083-a394-40a61dc9ad85},
 created = {2020-10-30T10:12:15.273Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.273Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {We study the generic problem of the escape of a classical particle over a fluctuating barrier under the influence of non-Gaussian noise mimicking the effects of nonequilibrated bath. The model system is described by a Langevin equation with two independent noise sources, one of which stands for the dichotomous process and the other describes external driving by α-stable noise. Our attention focuses on the effect of the structure of stable noises on the mean escape time and on the phenomenon of resonant activation. Possible physical interpretation of the occurrence of Lévy noises and the relevance of the model for chemical kinetics is briefly discussed. © 2004 The American Physical Society.},
 bibtype = {article},
 author = {Dybiec, B. and Gudowska-Nowak, E.},
 doi = {10.1103/PhysRevE.69.016105},
 journal = {Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics},
 number = {1}
}

We study the generic problem of the escape of a classical particle over a fluctuating barrier under the influence of non-Gaussian noise mimicking the effects of nonequilibrated bath. The model system is described by a Langevin equation with two independent noise sources, one of which stands for the dichotomous process and the other describes external driving by α-stable noise. Our attention focuses on the effect of the structure of stable noises on the mean escape time and on the phenomenon of resonant activation. Possible physical interpretation of the occurrence of Lévy noises and the relevance of the model for chemical kinetics is briefly discussed. © 2004 The American Physical Society.

@inproceedings{
 title = {Activation process driven by strongly non-Gaussian noises},
 type = {inproceedings},
 year = {2004},
 keywords = {Escape time,Numerical evaluation of the resonant activation,Resonant activation,Stable random variables,Stochastic resonance},
 volume = {5467},
 id = {614bb2c6-e468-34ff-895b-bc5167330019},
 created = {2020-10-30T10:12:15.779Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.779Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {The constructive role of non-Gaussian random fluctuations is studied in the context of the passage over the dichotomously switching potential barrier. Our attention focuses on the interplay of the effects of independent sources of fluctuations: an additive stable noise representing non-equilibrium external random force acting on the system and a fluctuating barrier. In particular, the influence of the structure of stable noises on the mean escape time and on the phenomenon of resonant activation (RA) is investigated. By use of the numerical Monte Carlo method it is documented that the suitable choice of the barrier switching rate and random external fields may produce resonant phenomenon leading to the enhancement of the kinetics and the shortest, most efficient reaction time.},
 bibtype = {inproceedings},
 author = {Dybiec, B. and Gudowska-Nowak, E.},
 doi = {10.1117/12.548564},
 booktitle = {Proceedings of SPIE - The International Society for Optical Engineering}
}

The constructive role of non-Gaussian random fluctuations is studied in the context of the passage over the dichotomously switching potential barrier. Our attention focuses on the interplay of the effects of independent sources of fluctuations: an additive stable noise representing non-equilibrium external random force acting on the system and a fluctuating barrier. In particular, the influence of the structure of stable noises on the mean escape time and on the phenomenon of resonant activation (RA) is investigated. By use of the numerical Monte Carlo method it is documented that the suitable choice of the barrier switching rate and random external fields may produce resonant phenomenon leading to the enhancement of the kinetics and the shortest, most efficient reaction time.

@article{
 title = {Resonant activation driven by strongly non-Gaussian noises},
 type = {article},
 year = {2004},
 keywords = {Mean first passage time,Non-Gaussian stable random variables,Resonant activation},
 volume = {4},
 id = {81d02b0b-471a-3ebf-9a5b-04b4b11ee5b9},
 created = {2020-10-30T10:12:15.977Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.977Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {The constructive role of non-Gaussian random fluctuations is studied in the context of the passage over the dichotomously switching potential barrier. Our attention focuses on the interplay of the effects of independent sources of fluctuations: an additive stable noise representing non-equilibrium external random force acting on the system and a fluctuating barrier. In particular, the influence of the structure of stable noises on the mean escape time and on the phenomenon of resonant activation (RA) is investigated. By use of the numerical Monte Carlo method it is documented that the suitable choice of the barrier switching rate and random external fields may produce resonant phenomenon leading to the enhancement of the kinetics and the shortest, most efficient reaction time.},
 bibtype = {article},
 author = {Dybiec, B. and Gudowska-Nowak, E.},
 doi = {10.1142/S0219477504001872},
 journal = {Fluctuation and Noise Letters},
 number = {2}
}

The constructive role of non-Gaussian random fluctuations is studied in the context of the passage over the dichotomously switching potential barrier. Our attention focuses on the interplay of the effects of independent sources of fluctuations: an additive stable noise representing non-equilibrium external random force acting on the system and a fluctuating barrier. In particular, the influence of the structure of stable noises on the mean escape time and on the phenomenon of resonant activation (RA) is investigated. By use of the numerical Monte Carlo method it is documented that the suitable choice of the barrier switching rate and random external fields may produce resonant phenomenon leading to the enhancement of the kinetics and the shortest, most efficient reaction time.

@article{
 title = {Resonant activation in the presence of nonequilibrated baths},
 type = {article},
 year = {2004},
 volume = {69},
 id = {7fd67483-63e8-3a37-be82-56530e0116c0},
 created = {2020-10-30T10:12:16.015Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.015Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {The study of the influence of non-Gaussian noise in a classical particle escape problem was discussed. The influence was studied using a model described by a Langevin equation with two independent noise sources. It took a nonequilibrated thermal bath which introduced a thermally activated process into consideration. The effect of the structure of stable noises on the mean escape time and on the phenomenon of resonant activation was also discussed.},
 bibtype = {article},
 author = {Dybiec, B. and Gudowska-Nowak, E.},
 journal = {Physical Review E - Statistical, Nonlinear, and Soft Matter Physics},
 number = {1 2}
}

The study of the influence of non-Gaussian noise in a classical particle escape problem was discussed. The influence was studied using a model described by a Langevin equation with two independent noise sources. It took a nonequilibrated thermal bath which introduced a thermally activated process into consideration. The effect of the structure of stable noises on the mean escape time and on the phenomenon of resonant activation was also discussed.

@article{
 title = {Controlling disease spread on networks with incomplete knowledge},
 type = {article},
 year = {2004},
 volume = {70},
 id = {b5196fb8-b0ef-39cb-911a-8b4ece06aab9},
 created = {2020-10-30T10:12:16.264Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.264Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {Models for control of highly infectious diseases on local, small-world, and scale-free networks are considered, with only partial information accessible about the status of individuals and their connections. We consider a case when individuals can be infectious before showing symptoms and thus before detection. For small to moderately severe incidence of infection with a small number of nonlocal links, it is possible to control disease spread by using purely local methods applied in a neighborhood centered around a detected infectious individual. There exists an optimal radius for such a control neighborhood leading to the lowest severity of the epidemic in terms of economic costs associated with disease and treatment. The efficiency of a local control strategy is very sensitive to the choice of the radius. Below the optimal radius, the local strategy is unsuccessful; the disease spreads throughout the system, necessitating treatment of the whole population. At the other extreme, a strategy involving a neighborhood that is too large controls the disease but is wasteful of resources. It is not possible to stop an epidemic on scale-free networks by preventive actions, unless a large proportion of the population is treated. © 2004 The American Physical Society.},
 bibtype = {article},
 author = {Dybiec, B. and Kleczkowski, A. and Gilligan, C.A.},
 doi = {10.1103/PhysRevE.70.066145},
 journal = {Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics},
 number = {6}
}

Models for control of highly infectious diseases on local, small-world, and scale-free networks are considered, with only partial information accessible about the status of individuals and their connections. We consider a case when individuals can be infectious before showing symptoms and thus before detection. For small to moderately severe incidence of infection with a small number of nonlocal links, it is possible to control disease spread by using purely local methods applied in a neighborhood centered around a detected infectious individual. There exists an optimal radius for such a control neighborhood leading to the lowest severity of the epidemic in terms of economic costs associated with disease and treatment. The efficiency of a local control strategy is very sensitive to the choice of the radius. Below the optimal radius, the local strategy is unsuccessful; the disease spreads throughout the system, necessitating treatment of the whole population. At the other extreme, a strategy involving a neighborhood that is too large controls the disease but is wasteful of resources. It is not possible to stop an epidemic on scale-free networks by preventive actions, unless a large proportion of the population is treated. © 2004 The American Physical Society.

@inproceedings{
 title = {Resonant effects in a voltage-activated channel gating},
 type = {inproceedings},
 year = {2004},
 keywords = {Channel gating,Hysteresis,Power spectrum,Resonant activation,Signal to noise ratio,Spectral amplification,Stochastic resonance,Synchronization},
 volume = {5467},
 id = {62a279a6-3dab-379f-9b6d-dcec7591909d},
 created = {2020-10-30T10:12:16.765Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.765Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {The non-selective voltage activated cation channel from the human red cells, which is activated at depolarizing potentials, has been shown to exhibit counter-clockwise gating hysteresis. We have analyzed the phenomenon with the simplest possible phenomenological models by assuming 2×2 discrete states, i.e. two normal open/closed states with two different states of "gate tension." Rates of transitions between the two branches of the hysteresis curve have been modeled with single-barrier kinetics by introducing a real-valued "reaction coordinate" parameterizing the protein's conformational change. When described in terms of the effective potential with cyclic variations of the control parameter (an activating voltage), this model exhibits typical "resonant effects": synchronization, resonant activation and stochastic resonance. Occurrence of the phenomena is investigated by running the stochastic dynamics of the model and analyzing statistical properties of gating trajectories.},
 bibtype = {inproceedings},
 author = {Gudowska-Nowak, E. and Dybiec, B. and Flyvbjerg, H.},
 doi = {10.1117/12.548556},
 booktitle = {Proceedings of SPIE - The International Society for Optical Engineering}
}

The non-selective voltage activated cation channel from the human red cells, which is activated at depolarizing potentials, has been shown to exhibit counter-clockwise gating hysteresis. We have analyzed the phenomenon with the simplest possible phenomenological models by assuming 2×2 discrete states, i.e. two normal open/closed states with two different states of "gate tension." Rates of transitions between the two branches of the hysteresis curve have been modeled with single-barrier kinetics by introducing a real-valued "reaction coordinate" parameterizing the protein's conformational change. When described in terms of the effective potential with cyclic variations of the control parameter (an activating voltage), this model exhibits typical "resonant effects": synchronization, resonant activation and stochastic resonance. Occurrence of the phenomena is investigated by running the stochastic dynamics of the model and analyzing statistical properties of gating trajectories.

@article{
 title = {Controlling disease spread on networks with incomplete knowledge},
 type = {article},
 year = {2004},
 volume = {70},
 id = {a97fed8a-9879-34d2-b5fd-471e677edb4a},
 created = {2020-10-30T10:12:16.814Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:16.814Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {The efficiency of local control strategies for stopping the spread of a disease on networks with a mixture of local and global links and cryptic infection was compared. A case when individual can be infectious before showing symptoms and thus before detection was considered. It was possible to control diseases spread by using purely local methods applied in a neighborhood centered around a detected infectious individual for small to moderately severe incidence of infection with a small number of nonlocal links. The results show that it is not possible to stop an epidemic on scale-free networks by preventive actions, unless a large proportion of the population is treated.},
 bibtype = {article},
 author = {Dybiec, B. and Kleczkowski, A. and Gilligan, C.A.},
 doi = {10.1103/PhysRevE.70.066145},
 journal = {Physical Review E - Statistical, Nonlinear, and Soft Matter Physics},
 number = {6 2}
}

The efficiency of local control strategies for stopping the spread of a disease on networks with a mixture of local and global links and cryptic infection was compared. A case when individual can be infectious before showing symptoms and thus before detection was considered. It was possible to control diseases spread by using purely local methods applied in a neighborhood centered around a detected infectious individual for small to moderately severe incidence of infection with a small number of nonlocal links. The results show that it is not possible to stop an epidemic on scale-free networks by preventive actions, unless a large proportion of the population is treated.

@article{
 title = {A toy model of faith-based systems evolution},
 type = {article},
 year = {2003},
 keywords = {Agent-based modeling,Lattice models,Power laws,Social dynamic,Social impact theory,Sociophysics},
 volume = {323},
 id = {31d960af-e883-3d63-8531-e659bf0b225c},
 created = {2020-10-30T10:12:17.020Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:17.020Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {A simple agent-based model of the evolution of faith-based systems (FBS) in human social networks is presented. In the model, each agent subscribes to a single FBS, and may be converted to share a different agent's FBS during social interactions. FBSs and agents each possess heritable quantitative traits that affect the probability of transmission of FBSs. The influence of social network conditions on the intermediate and final macroscopic states is examined. © 2003 Elsevier Science B.V. All rights reserved.},
 bibtype = {article},
 author = {Sadedin, S. and Dybiec, B. and Briscoe, G.},
 doi = {10.1016/S0378-4371(03)00046-3},
 journal = {Physica A: Statistical Mechanics and its Applications}
}

A simple agent-based model of the evolution of faith-based systems (FBS) in human social networks is presented. In the model, each agent subscribes to a single FBS, and may be converted to share a different agent's FBS during social interactions. FBSs and agents each possess heritable quantitative traits that affect the probability of transmission of FBSs. The influence of social network conditions on the intermediate and final macroscopic states is examined. © 2003 Elsevier Science B.V. All rights reserved.

@article{
 title = {Influence of the barrier shape on resonant activation},
 type = {article},
 year = {2002},
 volume = {66},
 id = {6546a601-38cb-3a14-b61c-5978b9b144d9},
 created = {2020-10-30T10:12:15.800Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:15.800Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {The escape of a Brownian particle over a dichotomously fluctuating barrier is investigated for various shapes of the barrier. The problem of resonant activation is revisited with the attention on the effect of the barrier shape on optimal value of the mean escape time in the system. The characteristic features of resonant behavior are analyzed for situations when the barrier switches either between different heights representing erection of a barrier and formation of a well, respectively, or it proceeds through “on” and “off” positions. © 2002 The American Physical Society.},
 bibtype = {article},
 author = {Dybiec, B. and Gudowska-Nowak, E.},
 doi = {10.1103/PhysRevE.66.026123},
 journal = {Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics},
 number = {2}
}

The escape of a Brownian particle over a dichotomously fluctuating barrier is investigated for various shapes of the barrier. The problem of resonant activation is revisited with the attention on the effect of the barrier shape on optimal value of the mean escape time in the system. The characteristic features of resonant behavior are analyzed for situations when the barrier switches either between different heights representing erection of a barrier and formation of a well, respectively, or it proceeds through “on” and “off” positions. © 2002 The American Physical Society.

@article{
 title = {Implication of barrier fluctuations on the rate of weakly adiabatic electron transfer},
 type = {article},
 year = {2002},
 keywords = {Escape time,Kinetic rate,Numerical evaluation of the resonant activation},
 volume = {13},
 id = {ef4b1177-4fe0-3266-8f11-9f154becbcde},
 created = {2020-10-30T10:12:17.092Z},
 file_attached = {false},
 profile_id = {f5390430-7317-381a-8c56-e25a878d78ef},
 last_modified = {2020-10-30T10:12:17.092Z},
 read = {false},
 starred = {false},
 authored = {true},
 confirmed = {false},
 hidden = {false},
 private_publication = {false},
 abstract = {The problem of escape of a Brownian particle in a cusp-shaped metastable potential is of special importance in nonadiabatic and weakly-adiabatic rate theory for electron transfer (ET) reactions. For the weakly-adiabatic reactions, the reaction follows an adiabaticity criterion in the presence of a sharp barrier. In contrast to the nonadiabatic case, the ET kinetics can be, however considerably influenced by the medium dynamics. In this paper, the problem of the escape time over a dichotomously fluctuating cusp barrier is discussed with its relevance to the high temperature ET reactions in condensed media.},
 bibtype = {article},
 author = {Dybiec, B. and Gudowska-Nowak, E. and Góra, P.F.},
 doi = {10.1142/S0129183102004078},
 journal = {International Journal of Modern Physics C},
 number = {9}
}

The problem of escape of a Brownian particle in a cusp-shaped metastable potential is of special importance in nonadiabatic and weakly-adiabatic rate theory for electron transfer (ET) reactions. For the weakly-adiabatic reactions, the reaction follows an adiabaticity criterion in the presence of a sharp barrier. In contrast to the nonadiabatic case, the ET kinetics can be, however considerably influenced by the medium dynamics. In this paper, the problem of the escape time over a dichotomously fluctuating cusp barrier is discussed with its relevance to the high temperature ET reactions in condensed media.