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Entropy 2017, 19(10), 511; https://doi.org/10.3390/e19100511

Far-From-Equilibrium Time Evolution between Two Gamma Distributions

1
School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UK
2
ENSTA ParisTech Université Paris-Saclay, 828 boulevard des Maréchaux, 91120 Palaiseau, France
3
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
4
DIMNP-UMR 5235 CNRS, Université de Montpellier, Place Eugène Bataillon, 34095 Montpellier, France
*
Author to whom correspondence should be addressed.
Received: 18 August 2017 / Revised: 18 September 2017 / Accepted: 21 September 2017 / Published: 22 September 2017
(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)
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Abstract

Many systems in nature and laboratories are far from equilibrium and exhibit significant fluctuations, invalidating the key assumptions of small fluctuations and short memory time in or near equilibrium. A full knowledge of Probability Distribution Functions (PDFs), especially time-dependent PDFs, becomes essential in understanding far-from-equilibrium processes. We consider a stochastic logistic model with multiplicative noise, which has gamma distributions as stationary PDFs. We numerically solve the transient relaxation problem and show that as the strength of the stochastic noise increases, the time-dependent PDFs increasingly deviate from gamma distributions. For sufficiently strong noise, a transition occurs whereby the PDF never reaches a stationary state, but instead, forms a peak that becomes ever more narrowly concentrated at the origin. The addition of an arbitrarily small amount of additive noise regularizes these solutions and re-establishes the existence of stationary solutions. In addition to diagnostic quantities such as mean value, standard deviation, skewness and kurtosis, the transitions between different solutions are analysed in terms of entropy and information length, the total number of statistically-distinguishable states that a system passes through in time. View Full-Text
Keywords: non-equilibrium statistical mechanics; gamma distribution; stochastic processes; Fokker-Planck equation; fluctuations and noise non-equilibrium statistical mechanics; gamma distribution; stochastic processes; Fokker-Planck equation; fluctuations and noise
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Kim, E.-J.; Tenkès, L.-M.; Hollerbach, R.; Radulescu, O. Far-From-Equilibrium Time Evolution between Two Gamma Distributions. Entropy 2017, 19, 511.

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