The distribution of forces affects vibrational properties in hard sphere glasses

The distribution of forces affects vibrational properties in hard sphere glasses. DeGiuli, E.; Lerner, E.; Brito, C.; and Wyart, M. arXiv, 2(1):1-8, 2014.

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We study how the distribution of contact forces, known to behave at small forces as $P(f)\sim f^\theta_f$, affects the stability and the vibrational properties of hard sphere glasses. As the jamming transition is approached we predict (i) the density of states $D(\omega)\sim \omega^a$ where $a=(\theta_f-1)/(3+\theta_f)$ and $\omega$ is the frequency, (ii) the shear modulus $\mu\sim (\phi_c-\phi)^-b $ where $b=(4+2\theta_f)/(3+\theta_f)$ and $\phi$ is the packing fraction and (iii) the mean square displacement $\langle \delta R^2\rangle\sim (\phi_c-\phi)^\kappa$, where $\kappa=2-2/(3+\theta_f)$. We test numerically (i) and provide data supporting that $\theta_f\approx 0.17$ independently of the system preparation in two and three dimensions, leading to $\kappa\approx1.37, a \approx -0.26$, and $b \approx 1.37$. Relation (iii) was previously unnoticed but appears to be satisfied in a recent replica calculation in infinite dimension, supporting that this approach captures some vibrational effects very precisely. However, our analysis supports that small and infinite dimension behave differently.

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 title = {The distribution of forces affects vibrational properties in hard sphere glasses},
 type = {article},
 year = {2014},
 identifiers = {[object Object]},
 keywords = {10003,4 washington place,brito 2,c,center for soft matter,degiuli 1,distribution of forces affects,e,lerner 1,m,new york,new york university,ny,research,sphere glasses,usa and,vibrational properties in hard,wyart 1},
 pages = {1-8},
 volume = {2},
 websites = {http://arxiv.org/abs/1402.3834},
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 abstract = {We study how the distribution of contact forces, known to behave at small forces as $P(f)\sim f^\theta_f$, affects the stability and the vibrational properties of hard sphere glasses. As the jamming transition is approached we predict (i) the density of states $D(\omega)\sim \omega^a$ where $a=(\theta_f-1)/(3+\theta_f)$ and $\omega$ is the frequency, (ii) the shear modulus $\mu\sim (\phi_c-\phi)^-b $ where $b=(4+2\theta_f)/(3+\theta_f)$ and $\phi$ is the packing fraction and (iii) the mean square displacement $\langle \delta R^2\rangle\sim (\phi_c-\phi)^\kappa$, where $\kappa=2-2/(3+\theta_f)$. We test numerically (i) and provide data supporting that $\theta_f\approx 0.17$ independently of the system preparation in two and three dimensions, leading to $\kappa\approx1.37, a \approx -0.26$, and $b \approx 1.37$. Relation (iii) was previously unnoticed but appears to be satisfied in a recent replica calculation in infinite dimension, supporting that this approach captures some vibrational effects very precisely. However, our analysis supports that small and infinite dimension behave differently.},
 bibtype = {article},
 author = {DeGiuli, E. and Lerner, E and Brito, C and Wyart, M},
 journal = {arXiv},
 number = {1}
}

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