Large times off-equilibrium dynamics of a particle in a random potential

TL;DR

This paper investigates the long-time off-equilibrium behavior of a particle in a high-dimensional random potential, revealing two distinct dynamical regimes and deriving analytical solutions that describe aging and stationary states.

Contribution

It provides a comprehensive analytical framework for understanding off-equilibrium dynamics in high-dimensional random potentials, including the derivation of equations and solutions for aging and stationary regimes.

Findings

Identification of two asymptotic regimes: stationary and aging dynamics.

Analytical solutions for correlation and response functions in the aging regime.

Discovery of non-analytic scaling functions and ultrametricity crossover for long-range correlations.

Abstract

We study the off-equilibrium dynamics of a particle in a general $N$-dimensional random potential when $N \to \infty$. We demonstrate the existence of two asymptotic time regimes: {\it i.} stationary dynamics, {\it ii.} slow aging dynamics with violation of equilibrium theorems. We derive the equations obeyed by the slowly varying part of the two-times correlation and response functions and obtain an analytical solution of these equations. For short-range correlated potentials we find that: {\it i.} the scaling function is non analytic at similar times and this behaviour crosses over to ultrametricity when the correlations become long range, {\it ii.} aging dynamics persists in the limit of zero confining mass with universal features for widely separated times. We compare with the numerical solution to the dynamical equations and generalize the dynamical equations to finite $N$ by extending the variational method to the dynamics.