Memory effects in the Fermi-Pasta-Ulam Model

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.

References in corpus (8)

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• Fermi, Pasta, Ulam and a mysterious lady
• Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals
• On the "generalized Generalized Langevin Equation"
• Proof of Nishida's conjecture on anharmonic lattices
• An averaging theorem for FPU in the thermodynamic limit
• Delocalization-Localization Transition due to Anharmonicity
• A series expansion for the time autocorrelation of dynamical variables

Cited by in corpus (8)

• Coarse-Grained Modelling Out of Equilibrium
• Power functional theory for many-body dynamics
• Jerky active matter: a phase field crystal model with translational and orientational memory
• On detailed balance in nonadiabatic dynamics: From spin spheres to equilibrium ellipsoids
• Generalized Langevin equations for systems with local interactions
• Structural localization in the Classical and Quantum Fermi-Pasta-Ulam Model
• Connection among stochastic Hamilton-Jacobi-Bellman equation, path-integral, and Koopman operator on nonlinear stochastic optimal control
• Statistics for stochastic differential equations and approximations of resolvent