Elastic Sheets, Phase Surfaces, and Pattern Universes

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 parallels between asymptotic expansions for the energy of elastic surfaces in powers of the thickness h and the free energy for almost periodic patterns expanded in powers of ε, the inverse aspect ratio of the pattern field. For sheets as well as patterns, the resulting energy can be expressed in terms of natural geometric invariants, the first and second fundamental forms of the elastic surface, respectively, the phase surface. We discuss various results for these energies and also address some of the outstanding questions. We extend previous work on point (in two dimensional) and loop (in three dimensional) disclinations and connect their topological indices with the condensation of Gaussian curvature of the phase surface. Motivated by this connection with the charge and spin of pattern quarks and leptons, we lay out an ambitious program to build a multiscale universe inspired by patterns in which the short (spatial and temporal) scales are given by a nearly periodic microstructure and whose macroscopic/slowly varying/averaged behaviors lead to a hierarchy of structures and features on much longer scales including analogs to quarks and leptons, dark matter, dark energy, and inflationary cosmology. One of our new findings is an interpretation of dark matter as the energy density in a pattern field. The associated gravitational forces naturally result in galactic rotation curves that are consistent with observations, while simultaneously avoiding some of the small‐scale difficulties of the standard ΛCDM (cold dark matter) paradigm in cosmology.

An SBV relaxation of the Cross-Newell energy for modeling stripe patterns and their defects

“…Rather, layers can bend and the spacing between layers can be nonuniform corresponding, respectively, to bending and stretching deformations of an ideal pattern. Stretching and bending deformations are 'topologically nice' and indeed there are useful analogies with bending and stretching deformations of this elastic sheets, that have been explored elsewhere [43]. In the context of this work, stretching and bending deformations break the global symmetries of an ideal stripe pattern, but the symmetry under translations along the stripes is still manifested locally.…”

“…These equations support the formation of shocks, so they need to be regularized by higher order effects in the small parameter [41,25]. An alternative to employing the Fredholm alternative/solvability is to directly compute an effective energy E[k(x, t)] by averaging the energy (2) over all the microstates that are consistent with a given macroscopic field k(x, t) [43]. This is equivalent to averaging over the phase shift θ 0 ∈ [0, 2π], and yields the energy functional ( 7)…”

“…where δ is 'small' and, ideally, we would want to take the limit δ → 0 + . This procedure is entirely analogous to the addition of the bending energy with a small coefficient to regularize unphysical behavior in the 'bare' Cross-Newell energy functional [41,43] to obtain the regularized Cross-Newell energy (7). As we will argue below, because of a regularizing effect akin to numerical viscosity, our method to solve the variational problem for the functional in (12) actually ends up solving the variational problem for the regularized functional (13) with a small δ > 0.…”

Pattern dark matter and galaxy scaling relations

“…where L P is the pattern Lagrangian density, d 4 x has dimensions of a length to the 4th power, the dimensionless metric g αβ has signature (− + + +) and ∇ μ is the corresponding covariant derivative. Equation (12) gives the natural covariant generalization [31] of the universal averaged energy for nearly periodic stripe patterns [30], and is thus expected to describe the macro-scopic behavior of phase hyper-surfaces in curved spacetimes for a variety of microscopic models [31]. The phase ψ is dimensionless, k μ := ∇ μ ψ has dimensions of inverse length, Σ * is a surface mass density scale, k 0 the preferred wavenumber, c the velocity of light so that Σ * c 2 /k 3 0 is a normalizing constant to ensure that S P has the correct dimensions for an action (the spacetime integral of an energy per unit volume).…”

Preface: Special Issues in Memory of David Benney (Part II)

“…In , Hitzazis and Fokas extended the unified transform method to the case of the Laplace, modified Helmholtz, and Helmholtz equations, formulated in a three‐dimensional cylindrical domain with a polygonal cross section. In , Newell and Venkataramani connected the theories of the deformation of elastic surfaces and phase surfaces arising in the description of almost periodic patterns, and built a multiscale universe inspired by patterns in which the short spatial and temporal scales are given by a nearly periodic microstructure.…”