Memory effects in the Fermi-Pasta-Ulam Model

TL;DR

This paper investigates the memory effects in the Fermi-Pasta-Ulam Model's dynamics at thermal equilibrium, combining numerical simulations and analytical calculations to understand relaxation behaviors across temperature regimes.

Contribution

It introduces an analytical method to compute high-order Taylor coefficients of the ISF and links the dynamics to a memory-dependent GLE, revealing new insights into the model's relaxation processes.

Findings

At high temperatures, the system behaves as an ideal gas.

At low excitations, the system acts as a harmonic chain.

Intermediate temperatures show nontrivial ISF relaxation.

Abstract

We study the Intermediate Scattering Function (ISF) of the strongly-nonlinear Fermi-Pasta Ulam Model at thermal equilibrium, using both numerical and analytical methods. From the molecular dynamics simulations we distinguish two limit regimes, as the system behaves as an ideal gas at high temperature and as a harmonic chain for low excitations. At intermediate temperatures the ISF relaxes to equilibrium in a nontrivial fashion. We then calculate analytically the Taylor coefficients of the ISF to arbitrarily high orders (the specific, simple shape of the two-body interaction allows us to derive an iterative scheme for these.) The results of the recursion are in good agreement with the numerical ones. Via an estimate of the complete series expansion of the scattering function, we can reconstruct within a certain temperature range its coarse-grained dynamics. This is governed by a memory-dependent Generalized Langevin Equation (GLE), which can be derived via projection operator techniques. Moreover, by analyzing the first series coefficients of the ISF, we can extract a parameter associated to the strength of the memory effects in the dynamics.