The Structure of Fluctuating Thin Sheets Under Random Forcing

TL;DR

This paper develops a mathematical model combining elasticity and statistical physics to describe the fluctuations of thin sheets under random forces, revealing a logarithmically corrected structure factor rather than a power-law.

Contribution

It introduces a novel nonlinear Langevin equation for fluctuating thin sheets and provides an analytical solution for the structure factor using the self-consistent expansion method.

Findings

The structure factor follows a logarithmically corrected rational function.

Numerical simulations confirm the analytical predictions.

Contrasts with previous suggestions of a power-law behavior.

Abstract

We propose a mathematical model to describe the athermal fluctuations of thin sheets driven by the type of random driving that might be experienced prior to weak crumpling. The model is obtained by merging the F\"oppl-von K\'arm\'an equations from elasticity theory with techniques from out-of-equilibrium statistical physics to obtain a nonlinear strongly coupled $\phi^{4}$-Langevin field equation with spatially varying kernel. With the aid of the self-consistent expansion (SCE), this equation is analytically solved for the structure factor of a fluctuating sheet. In contrast to previous research which has suggested that the structure factor follows an anomalous power-law, we find that the structure factor in fact obeys a logarithmically corrected rational function. Numerical simulations of our model confirm the accuracy of our analytical solution.