Non-ergodic transitions in many-body Langevin systems: a method of dynamical system reduction

TL;DR

This paper investigates non-ergodic transitions in many-body Langevin systems by deriving a correlation function equation, identifying a critical temperature, and reducing the dynamics to a two-dimensional system to analyze transition behavior.

Contribution

It introduces a dynamical system reduction method to analyze non-ergodic transitions in many-body Langevin systems near critical temperatures.

Findings

Identification of a critical temperature where system behavior changes.

Reduction of complex equations to a 2D ODE system.

Demonstration of non-ergodic transition occurring above the critical temperature.

Abstract

We study a non-ergodic transition in a many-body Langevin system. We first derive an equation for the two-point time correlation function of density fluctuations, ignoring the contributions of the third- and fourth-order cumulants. For this equation, with the average density fixed, we find that there is a critical temperature at which the qualitative nature of the trajectories around the trivial solution changes. Using a method of dynamical system reduction around the critical temperature, we simplify the equation for the time correlation function into a two-dimensional ordinary differential equation. Analyzing this differential equation, we demonstrate that a non-ergodic transition occurs at some temperature slightly higher than the critical temperature.