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Probability and Combinatorics

Tuesday, October 10, 2023 - 3:30pm

Joshua McGinnis

University of Pennsylvania


Temple University

Wachman Hall 617

We review recent results regarding the rigorous approximation of 1D and 2D disordered (random, independent masses and/or springs) harmonic lattices by effective wave equations in the long wave limit. In this linear setting, we show the homogenization argument and highlight the tools used from probability theory to control the stochastic error terms such as the Law of the Iterated Logarithm and Hoeffding’s inequality. With our discussion of the linear problem serving as a springboard, we then present a new result regarding the approximation of an FPUT lattice with random masses by a KdV equation. Specifically, we are able to bound the approximation error in terms of the small parameter from the long wave scaling in an almost sure sense. In our theorem, we require a technical condition on the random masses, which we call transparency. Our proof relies on the incorporation of an auto-regressive process into an approximating ansatz, which itself is approximated by solutions to the KdV equation. We discuss the role of the auto-regressive process as well as the condition of transparency in the proof and give numerical evidence supporting the result. We conclude by discussing open questions such as the apparent lack of KdV dynamics in an FPUT lattice with independent, random masses.