dorsal/arxiv
View SchemaA nonlinear model dynamics for closed-system, constrained, maximal-entropy-generation relaxation by energy redistribution
| Authors | Gian Paolo Beretta |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0501178 |
| URL | https://arxiv.org/abs/quant-ph/0501178 |
| DOI | 10.1103/PhysRevE.73.026113 |
| Journal | Physical Review E, Vol.73, 026113 (2006) |
Abstract
We discuss a nonlinear model for the relaxation by energy redistribution within an isolated, closed system composed of non-interacting identical particles with energy levels e_i with i=1,2,...,N. The time-dependent occupation probabilities p_i(t) are assumed to obey the nonlinear rate equations tau dp_i/dt=-p_i ln p_i+ alpha(t)p_i-beta(t)e_ip_i where alpha(t) and beta(t) are functionals of the p_i(t)'s that maintain invariant the mean energy E=sum_i e_ip_i(t) and the normalization condition 1=sum_i p_i(t). The entropy S(t)=-k sum_i p_i(t) ln p_i(t) is a non-decreasing 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 p_i(t) of the rate equations are unique and well-defined for arbitrary initial conditions p_i(0) and for all times. Existence and uniqueness both forward and backward in time allows the reconstruction of the primordial lowest entropy state. The 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 of the nonlinear dynamical equation proposed in a series of papers ended with G.P.Beretta, Found.Phys., 17, 365 (1987) and recently rediscovered in S. Gheorghiu-Svirschevski, Phys.Rev.A, 63, 022105 and 054102 (2001). Numerical results illustrate the features of the dynamics and the differences with the rate equations recently considered for the same problem in M.Lemanska and Z.Jaeger, Physica D, 170, 72 (2002).
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"abstract": "We discuss a nonlinear model for the relaxation by energy redistribution\nwithin an isolated, closed system composed of non-interacting identical\nparticles with energy levels e_i with i=1,2,...,N. The time-dependent\noccupation probabilities p_i(t) are assumed to obey the nonlinear rate\nequations tau dp_i/dt=-p_i ln p_i+ alpha(t)p_i-beta(t)e_ip_i where alpha(t) and\nbeta(t) are functionals of the p_i(t)\u0027s that maintain invariant the mean energy\nE=sum_i e_ip_i(t) and the normalization condition 1=sum_i p_i(t). The entropy\nS(t)=-k sum_i p_i(t) ln p_i(t) is a non-decreasing function of time until the\ninitially nonzero occupation probabilities reach a Boltzmann-like canonical\ndistribution over the occupied energy eigenstates. Initially zero occupation\nprobabilities, instead, remain zero at all times. The solutions p_i(t) of the\nrate equations are unique and well-defined for arbitrary initial conditions\np_i(0) and for all times. Existence and uniqueness both forward and backward in\ntime allows the reconstruction of the primordial lowest entropy state. The time\nevolution is at all times along the local direction of steepest entropy ascent\nor, equivalently, of maximal entropy generation. These rate equations have the\nsame mathematical structure and basic features of the nonlinear dynamical\nequation proposed in a series of papers ended with G.P.Beretta, Found.Phys.,\n17, 365 (1987) and recently rediscovered in S. Gheorghiu-Svirschevski,\nPhys.Rev.A, 63, 022105 and 054102 (2001). Numerical results illustrate the\nfeatures of the dynamics and the differences with the rate equations recently\nconsidered for the same problem in M.Lemanska and Z.Jaeger, Physica D, 170, 72\n(2002).",
"arxiv_id": "quant-ph/0501178",
"authors": [
"Gian Paolo Beretta"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevE.73.026113",
"journal_ref": "Physical Review E, Vol.73, 026113 (2006)",
"title": "A nonlinear model dynamics for closed-system, constrained, maximal-entropy-generation relaxation by energy redistribution",
"url": "https://arxiv.org/abs/quant-ph/0501178"
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