Local equilibrium properties of ultraslow diffusion in the Sinai model

TL;DR

This study investigates the equilibrium-like behavior of ultraslow diffusion in the Sinai model, revealing how particles exhibit local equilibrium properties despite overall non-equilibrium dynamics, through extensive numerical simulations.

Contribution

It demonstrates that unbounded Sinai diffusion closely resembles equilibrium states of finite systems due to time scale separation, supported by large-scale numerical analysis.

Findings

Unbounded motion approximates finite system equilibrium at all times.

Fast local equilibration within wells contrasts with slow barrier crossing.

Finite-time crossover phenomena are observed up to very long times.

Abstract

We perform numerical studies of a thermally driven, overdamped particle in a random quenched force field, known as the Sinai model. We compare the unbounded motion on an infinite 1-dimensional domain to the motion in bounded domains with reflecting boundaries and show that the unbounded motion is at every time close to the equilibrium state of a finite system of growing size. This is due to time scale separation: Inside wells of the random potential, there is relatively fast equilibration, while the motion across major potential barriers is ultraslow. Quantities studied by us are the time dependent mean squared displacement, the time dependent mean energy of an ensemble of particles, and the time dependent entropy of the probability distribution. Using a very fast numerical algorithm, we can explore times up top $10^{17}$ steps and thereby also study finite-time crossover phenomena.