Non-normalizable quasi-equilibrium solution of the Fokker-Planck equation for nonconfining fields
| Autor: | Eli Barkai, Celia Anteneodo, David A. Kessler, Lucianno Defaveri |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
FOS: Physical sciences
General Physics and Astronomy lcsh:Astrophysics Boltzmann-Gibbs regularization 01 natural sciences Article 010305 fluids & plasmas 0103 physical sciences lcsh:QB460-466 Boundary value problem Statistical physics 010306 general physics lcsh:Science Condensed Matter - Statistical Mechanics Physics Partition function (statistical mechanics) Statistical Mechanics (cond-mat.stat-mech) non-confining fields Observable Eigenfunction lcsh:QC1-999 Mean squared displacement quasi-equilibrium Error function Regularization (physics) Fokker–Planck equation lcsh:Q lcsh:Physics |
| Zdroj: | Entropy Volume 23 Issue 2 Entropy, Vol 23, Iss 131, p 131 (2021) |
| Popis: | We investigate the overdamped Langevin motion for particles in a potential well that is asymptotically flat. When the potential well is deep as compared to the temperature, physical observables, like the mean square displacement, are essentially time-independent over a long time interval, the stagnation epoch. However, the standard Boltzmann&ndash 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 for 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&ndash Planck equation with free boundary conditions, we obtain an approximate time-independent solution of the BG form, being valid for times that are long, but still short as compared to the exponentially large escape time. The escaped particles follow a general free-particle statistics, where the solution is an error function, which is shifted due to the initial struggle to overcome the potential well. With the eigenfunction solution of the Fokker&ndash 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. |
| Databáze: | OpenAIRE |
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