Abstract
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–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–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–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.
Original language | English |
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Article number | 131 |
Pages (from-to) | 1-16 |
Number of pages | 16 |
Journal | Entropy |
Volume | 23 |
Issue number | 2 |
DOIs | |
State | Published - 20 Jan 2021 |
Bibliographical note
Publisher Copyright:© 2021 by the authors. Licensee MDPI, Basel, Switzerland.
Funding
Funding: D.A.K. and E.B. acknowledge the support of Israel Science Foundation’s grant 1898/17. C.A. acknowledges partial financial support from Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ). C.A. and L.D. acknowledge support from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brazil (CAPES)—Finance Code 001.
Funders | Funder number |
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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior | |
Conselho Nacional de Desenvolvimento Científico e Tecnológico | |
Israel Science Foundation | 1898/17 |
Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro |
Keywords
- Boltzmann-Gibbs regularization
- Non-confining fields
- Quasi-equilibrium