Thermally driven elastic membranes are quasi-linear across all scales

Abstract

We study the static and dynamic structure of thermally fluctuating elastic thin sheets by investigating a model known as the overdamped dynamic Föppl-von Kármán equation, in which the Föppl-von Kármán equation from elasticity theory is driven by white noise. The resulting nonlinear equation is governed by a single nondimensional coupling parameter g where large and small values of g correspond to weak and strong nonlinear coupling respectively. By analysing the weak coupling case with ordinary perturbation theory and the strong coupling case with a self-consistent methodology known as the self-consistent expansion, precise analytic predictions for the static and dynamic structure factors are obtained. Importantly, the maximum frequency n max supported by the system plays a role in determining which of three possible classes such sheets belong to: (1) when g ≫ 1, the system is mostly linear with roughness exponent ζ = 1 and dynamic exponent z = 4, (2) when g ≪ 2/n max, the system is extremely nonlinear with roughness exponent ζ = 1/2 and dynamic exponent z = 3, (3) between these regimes, an intermediate behaviour is obtained in which a crossover occurs such that the nonlinear behaviour is observed for small frequencies while the linear behaviour is observed for large frequencies, and thus the large frequency linear tail is found to have a significant impact on the small frequency behaviour of the sheet. Back-of-the-envelope calculations suggest that ultra-thin materials such as graphene lie in this intermediate regime. Despite the existence of these three distinct behaviours, the decay rate of the dynamic structure factor is related to the static structure factor as if the system were completely linear. This quasi-linearity occurs regardless of the size of g and at all length scales. Numerical simulations confirm the existence of the three classes of behaviour and the quasi-linearity of all classes.

Asymptotic matching the self-consistent expansion to approximate the modified Bessel functions of the second kind

“…The self-consistent expansion (SCE) is a powerful technique in statistical physics for obtaining perturbative expansions of many-body interacting systems. First developed by Schwartz and Edwards to investigate the KPZ equation for surface growth [10], it has since been applied to a number of problems in statistical physics, including generalisations of the KPZ equation [11][12][13][14][15][16][17][18][19], turbulence [20] , wetting fronts and fracture [21][22][23][24], the XY-model [25] and fluctuating elastic sheets [26][27][28]. The basic idea of the method is that when performing a perturbative expansion of any particular system, one will always have various degrees of freedom in the selection of the zeroth order system.…”