Non-equilibrium attractor for non-linear stochastic dynamics

¹ Física Teórica, Universidad de Sevilla - Apartado de Correos 1065, E-41080 Sevilla, Spain
² Departamento de Física Aplicada I, E.P.S., Universidad de Sevilla - E-41011 Sevilla, Spain
³ LPTMS, Université Paris-Saclay, CNRS - F-91405, Orsay, France
⁴ Ecole Normale Supérieure de Lyon - F-69364 Lyon, France

Received: 15 September 2023
Accepted: 11 January 2024

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.