Arxiv Journal doi abstract bibtex

We connect the theories of the deformation of elastic surfaces and phase surfaces arising in the description of almost periodic patterns. In particular, we show striking parallels between expansions for the energy of elastic surfaces in powers of the thickness h and the free energy averaged over a period of an almost periodic pattern expanded in powers of ϵ, the inverse aspect ratio of the pattern field. In both cases, the resulting energy can be expressed in terms of the first and second fundamental forms of the surfaces involved, the elastic surface in the former case and the phase surface in the latter. We discuss various results that are obtained by exploiting this analogy and also address some of the outstanding questions. One common result of particular interest concerns the condensation of the Gaussian curvature onto isolated point defects in two dimensions and onto loop filaments in three dimensions. We also lay out an ambitious and somewhat speculative program to build a multi-scale model of the universe inspired by patterns, in which the short (spatial and temporal) scale structure on the Planck scales is given by a nearly periodic microstructure, and macroscopic/slowly varying/averaged behaviors on scales much larger than the Planck scale leads to a hierarchy of structures and features including analogues of quarks, leptons, dark matter, dark energy and inflationary cosmology.

@article{pattern-universe, Abstract = {We connect the theories of the deformation of elastic surfaces and phase surfaces arising in the description of almost periodic patterns. In particular, we show striking parallels between expansions for the energy of elastic surfaces in powers of the thickness h and the free energy averaged over a period of an almost periodic pattern expanded in powers of ϵ, the inverse aspect ratio of the pattern field. In both cases, the resulting energy can be expressed in terms of the first and second fundamental forms of the surfaces involved, the elastic surface in the former case and the phase surface in the latter. We discuss various results that are obtained by exploiting this analogy and also address some of the outstanding questions. One common result of particular interest concerns the condensation of the Gaussian curvature onto isolated point defects in two dimensions and onto loop filaments in three dimensions. We also lay out an ambitious and somewhat speculative program to build a multi-scale model of the universe inspired by patterns, in which the short (spatial and temporal) scale structure on the Planck scales is given by a nearly periodic microstructure, and macroscopic/slowly varying/averaged behaviors on scales much larger than the Planck scale leads to a hierarchy of structures and features including analogues of quarks, leptons, dark matter, dark energy and inflationary cosmology. }, Author = {Alan C. Newell and Shankar C. Venkataramani}, Date-Added = {2017-10-15 18:02:26 +0000}, Date-Modified = {2017-10-15 18:15:10 +0000}, Doi = {http://onlinelibrary.wiley.com/doi/10.1111/sapm.12184/full}, Journal = {Studies in Applied Mathematics}, Keywords = {pubs}, Number = {2}, Pages = {322--368}, Title = {Elastic sheets, phase surfaces and pattern universes}, Url_Arxiv = {arxiv.org/abs/1612.01007}, Url_Journal = {http://onlinelibrary.wiley.com/doi/10.1111/sapm.12184/full}, Volume = {139}, Year = {2017}, Bdsk-Url-1 = {http://onlinelibrary.wiley.com/doi/10.1111/sapm.12184/full}}

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