Non-normalizable quasi-equilibrium solution of the Fokker-Planck equation for nonconfining fields

TL;DR

This paper studies the non-normalizable quasi-equilibrium states of particles in non-confining potentials, showing how regularized Boltzmann-Gibbs statistics accurately describe long-time behavior despite divergence issues.

Contribution

It derives an approximate time-independent solution of the Fokker-Planck equation for non-confining fields, validating the use of regularized BG statistics in non-normalizable regimes.

Findings

The eigenfunction expansion yields a valid quasi-stationary solution.

Regularized BG statistics accurately predict observables in the long-time regime.

The solution describes particles escaping with free-particle statistics.

Abstract

We investigate the overdamped Langevin motion for particles in a potential well that is asymptotically flat. When the potential well is deep compared to temperature, physical observables like the mean square displacement are essentially time-independent over a long time interval, the stagnation epoch. However the standard Boltzmann-Gibbs (BG) distribution is non-normalizable, given that the usual partition function is divergent. For this regime, we have previously shown that a regularization of BG statistics allows the prediction of the values of dynamical and thermodynamical observables in the non-normalizable quasi-equilibrium state. In this work, based on the eigenfunction expansion of the time-dependent solution of the associated Fokker-Planck equation with free boundary conditions, we obtain an approximate time-independent solution of the BG form, valid for times which are long, but still short compared to the exponentially large escape time. The escaped particles follow a general free-particle statistics, where the solution is a an error function, shifted due to the initial struggle to overcome the potential well. With the eigenfunction solution of the Fokker-Planck equation in hand, we show the validity of the regularized BG statistics and how it perfectly describes the time-independent regime though the quasi-stationary state is non-normalizable.