Out-of-equilibrium dynamical equations of infinite-dimensional particle systems. I. The isotropic case

TL;DR

This paper derives exact out-of-equilibrium dynamical equations for infinite-dimensional particle systems, applicable to various physical scenarios, and introduces a single effective stochastic equation to analyze complex many-body dynamics.

Contribution

It provides the first exact reduction of high-dimensional particle dynamics to a one-dimensional stochastic equation in the infinite-dimensional limit, applicable to diverse out-of-equilibrium systems.

Findings

Exact dynamical equations for infinite-dimensional systems

Effective stochastic equation with self-consistent kernels

Application to state-following equations for glasses

Abstract

We consider the Langevin dynamics of a many-body system of interacting particles in $d$ dimensions, in a very general setting suitable to model several out-of-equilibrium situations, such as liquid and glass rheology, active self-propelled particles, and glassy aging dynamics. The pair interaction potential is generic, and can be chosen to model colloids, atomic liquids, and granular materials. In the limit ${d\to\infty}$, we show that the dynamics can be exactly reduced to a single one-dimensional effective stochastic equation, with an effective thermal bath described by kernels that have to be determined self-consistently. We present two complementary derivations, via a dynamical cavity method and via a path-integral approach. From the effective stochastic equation, one can compute dynamical observables such as pressure, shear stress, particle mean-square displacement, and the associated response function. As an application of our results, we derive dynamically the `state-following' equations that describe the response of a glass to quasistatic perturbations, thus bypassing the use of replicas. The article is written in a modular way, that allows the reader to skip the details of the derivations and focus on the physical setting and the main results.