# Generalized Langevin equations for systems with local interactions

## TL;DR

This paper introduces a novel method to approximate the memory integral in generalized Langevin equations for high-dimensional nonlinear systems with local interactions, enabling accurate computation of statistical properties.

## Contribution

It develops a new approximation technique for the Mori-Zwanzig memory integral using Faber operator series and an exact combinatorial algorithm, advancing the analysis of complex systems.

## Key findings

- Effective computation of auto-correlation functions demonstrated
- Accurate modeling of intermediate scattering functions achieved
- New stochastic process representation for equilibrium systems

## Abstract

We present a new method to approximate the Mori-Zwanzig (MZ) memory integral in generalized Langevin equations (GLEs) describing the evolution of smooth observables in high-dimensional nonlinear systems with local interactions. Building upon the Faber operator series we recently developed for the orthogonal dynamics propagator, and an exact combinatorial algorithm that allows us to compute memory kernels from first principles, we demonstrate that the proposed method is effective in computing auto-correlation functions, intermediate scattering functions and other important statistical properties of the observable. We also develop a new stochastic process representation of the MZ fluctuation term for systems in statistical equilibrium. Numerical applications are presented for the Fermi-Pasta-Ulam model, and for random wave propagation in homogeneous media.