Non-equilibrium attractor for non-linear stochastic dynamics

TL;DR

This paper investigates the emergence of a long-lived, non-equilibrium attractor state in mesoscopic stochastic systems after a temperature quench, revealing a universal behavior in both one-dimensional and higher-dimensional systems.

Contribution

It introduces the concept of a non-equilibrium attractor state in stochastic dynamics, supported by numerical evidence and physical arguments, extending to complex higher-dimensional systems.

Findings

A long-lived Dirac-delta distribution emerges after temperature quench.

The attractor state dominates dynamics over an intermediate timescale.

The phenomenon likely extends to anisotropic and interacting systems.

Abstract

We study the dynamical behaviour of mesoscopic systems in contact with a thermal bath, described either via a non-linear Langevin equation at the trajectory level -- or the corresponding Fokker-Planck equation for the probability distribution function at the ensemble level. Our focus is put on one-dimensional -- or $d$-dimensional isotropic -- systems in confining potentials, with detailed balance -- fluctuation-dissipation thus holds, and the stationary probability distribution has the canonical form at the bath temperature. When quenching the bath temperature to low enough values, a far-from-equilibrium state emerges that rules the dynamics over a characteristic intermediate timescale. Such a long-lived state has a Dirac-delta probability distribution function and attracts all solutions over this intermediate timescale, in which the initial conditions are immaterial while the influence of the bath is still negligible. Numerical evidence and qualitative physical arguments suggest that the above picture extends to higher-dimensional systems, with anisotropy and interactions.