Nonlinear model dynamics for closed-system, constrained, maximal-entropy-generation relaxation by energy redistribution

Gian Paolo Beretta
Phys. Rev. E 73, 026113 – Published 13 February 2006

Abstract

We discuss a nonlinear model for relaxation by energy redistribution within an isolated, closed system composed of noninteracting identical particles with energy levels ei with i=1,2,,N. The time-dependent occupation probabilities pi(t) are assumed to obey the nonlinear rate equations τdpidt=pilnpiα(t)piβ(t)eipi where α(t) and β(t) are functionals of the pi(t)’s that maintain invariant the mean energy E=i=1Neipi(t) and the normalization condition 1=i=1Npi(t). The entropy S(t)=kBi=1Npi(t)lnpi(t) is a nondecreasing function of time until the initially nonzero occupation probabilities reach a Boltzmann-like canonical distribution over the occupied energy eigenstates. Initially zero occupation probabilities, instead, remain zero at all times. The solutions pi(t) of the rate equations are unique and well defined for arbitrary initial conditions pi(0) and for all times. The existence and uniqueness both forward and backward in time allows the reconstruction of the ancestral or primordial lowest entropy state. By casting the rate equations in terms not of the pi’s but of their positive square roots pi, they unfold from the assumption that time evolution is at all times along the local direction of steepest entropy ascent or, equivalently, of maximal entropy generation. These rate equations have the same mathematical structure and basic features as the nonlinear dynamical equation proposed in a series of papers ending with G. P. Beretta, Found. Phys. 17, 365 (1987) and recently rediscovered by S. Gheorghiu-Svirschevski [Phys. Rev. A 63, 022105 (2001);63, 054102 (2001)]. Numerical results illustrate the features of the dynamics and the differences from the rate equations recently considered for the same problem by M. Lemanska and Z. Jaeger [Physica D 170, 72 (2002)]. We also interpret the functionals kBα(t) and kBβ(t) as nonequilibrium generalizations of the thermodynamic-equilibrium Massieu characteristic function and inverse temperature, respectively.

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  • Received 30 January 2005

DOI:https://doi.org/10.1103/PhysRevE.73.026113

©2006 American Physical Society

Authors & Affiliations

Gian Paolo Beretta*

  • Università di Brescia, via Branze 38, 25123 Brescia, Italy

  • *Electronic address: beretta@unibs.it

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Vol. 73, Iss. 2 — February 2006

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