Using Random Walks to Establish Wavelike Behavior in an FPUT System with Random Coefficients

TL;DR

This paper demonstrates that in a stochastic FPUT lattice with random coefficients, long wave solutions tend to behave like wave equations, with convergence proven using stochastic homogenization and random walk theory.

Contribution

It introduces a novel approach combining stochastic homogenization and random walk results to analyze wave-like behavior in a random FPUT system.

Findings

Solutions converge to wave equation solutions in a strong sense

Convergence is both almost sure and in expectation

The convergence rate is slow

Abstract

We consider a linear Fermi-Pasta-Ulam-Tsingou lattice with random spatially varying material coefficients. Using the methods of stochastic homogenization we show that solutions with long wave initial data converge in an appropriate sense to solutions of a wave equation. The convergence is strong and both almost sure and in expectation, but the rate is quite slow. The technique combines energy estimates with powerful classical results about random walks, specifically the law of the iterated logarithm.