Thermally driven elastic membranes are quasi-linear across all scales

We study the static and dynamic structure of thermally fluctuating elastic thin sheets by investigating the overdamped dynamic F\"oppl-von K\'arm\'an equation, in which the F\"oppl-von K\'arm\'an equation from elasticity theory is driven by white noise. This nonlinear equation is governed by a single nondimensional coupling parameter $g$ whose large and small values 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. 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 $g\gg1$, the system is mostly linear with roughness exponent $\zeta=1$ and dynamic exponent $z=4$, (2) when $g\ll2/n_{\max}$, the system is extremely nonlinear with roughness exponent $\zeta=1/2$ 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. 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.