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Non-Equilibrium Liouville and Wigner Equations: Classical Statistical Mechanics and Chemical Reactions for Long Times

Departamento de Fisica Teorica, Facultad de Ciencias Fisicas, Universidad Complutense, 28040 Madrid, Spain
Entropy 2019, 21(2), 179; https://doi.org/10.3390/e21020179
Received: 26 November 2018 / Revised: 18 January 2019 / Accepted: 7 February 2019 / Published: 14 February 2019
(This article belongs to the Special Issue 20th Anniversary of Entropy—Review Papers Collection)
We review and improve previous work on non-equilibrium classical and quantum statistical systems, subject to potentials, without ab initio dissipation. We treat classical closed three-dimensional many-particle interacting systems without any “heat bath” (h b), evolving through the Liouville equation for the non-equilibrium classical distribution W c, with initial states describing thermal equilibrium at large distances but non-equilibrium at finite distances. We use Boltzmann’s Gaussian classical equilibrium distribution W c , e q, as weight function to generate orthogonal polynomials (H n’s) in momenta. The moments of W c, implied by the H n’s, fulfill a non-equilibrium hierarchy. Under long-term approximations, the lowest moment dominates the evolution towards thermal equilibrium. A non-increasing Liapunov function characterizes the long-term evolution towards equilibrium. Non-equilibrium chemical reactions involving two and three particles in a h b are studied classically and quantum-mechanically (by using Wigner functions W). Difficulties related to the non-positivity of W are bypassed. Equilibrium Wigner functions W e q generate orthogonal polynomials, which yield non-equilibrium moments of W and hierarchies. In regimes typical of chemical reactions (short thermal wavelength and long times), non-equilibrium hierarchies yield approximate Smoluchowski-like equations displaying dissipation and quantum effects. The study of three-particle chemical reactions is new. View Full-Text
Keywords: non-equilibrium Liouville and Wigner distributions; equilibrium solutions and orthogonal polynomials; long-term irreversible approach of non-equilibrium moments to thermal equilibrium; chemical reactions for two and three particles non-equilibrium Liouville and Wigner distributions; equilibrium solutions and orthogonal polynomials; long-term irreversible approach of non-equilibrium moments to thermal equilibrium; chemical reactions for two and three particles
MDPI and ACS Style

Álvarez-Estrada, R.F. Non-Equilibrium Liouville and Wigner Equations: Classical Statistical Mechanics and Chemical Reactions for Long Times. Entropy 2019, 21, 179. https://doi.org/10.3390/e21020179

AMA Style

Álvarez-Estrada RF. Non-Equilibrium Liouville and Wigner Equations: Classical Statistical Mechanics and Chemical Reactions for Long Times. Entropy. 2019; 21(2):179. https://doi.org/10.3390/e21020179

Chicago/Turabian Style

Álvarez-Estrada, Ramon F. 2019. "Non-Equilibrium Liouville and Wigner Equations: Classical Statistical Mechanics and Chemical Reactions for Long Times" Entropy 21, no. 2: 179. https://doi.org/10.3390/e21020179

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