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Entropy 2014, 16(2), 885-894; doi:10.3390/e16020885

From Random Motion of Hamiltonian Systems to Boltzmann’s H Theorem and Second Law of Thermodynamics: a Pathway by Path Probability

1
Laboratoire de Physique Statistique et Systems Complexes, ISMANS, 44 Ave. F.A., Bartholdi, 72000 Le Mans, France
2
IMMM, Université du Maine, Ave. O. Messiaen, Le Mans 72085, France
*
Author to whom correspondence should be addressed.
Received: 25 November 2013 / Revised: 18 December 2013 / Accepted: 23 January 2014 / Published: 13 February 2014
(This article belongs to the collection Advances in Applied Statistical Mechanics)
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Abstract

A numerical experiment of ideal stochastic motion of a particle subject to conservative forces and Gaussian noise reveals that the path probability depends exponentially on action. This distribution implies a fundamental principle generalizing the least action principle of the Hamiltonian/Lagrangian mechanics and yields an extended formalism of mechanics for random dynamics. Within this theory, Liouville’s theorem of conservation of phase density distribution must be modified to allow time evolution of phase density and consequently the Boltzmann H theorem. We argue that the gap between the regular Newtonian dynamics and the random dynamics was not considered in the criticisms of the H theorem.
Keywords: statistical mechanics; random motion; path probability; least action statistical mechanics; random motion; path probability; least action
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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Wang, Q.A.; El Kaabouchiu, A. From Random Motion of Hamiltonian Systems to Boltzmann’s H Theorem and Second Law of Thermodynamics: a Pathway by Path Probability. Entropy 2014, 16, 885-894.

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