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” ( ), evolving through the Liouville equation for the non-equilibrium classical distribution , with initial states describing thermal equilibrium at large distances but non-equilibrium at finite distances. We use Boltzmann’s Gaussian classical equilibrium distribution , as weight function to generate orthogonal polynomials ( ’s) in momenta. The moments of , implied by the ’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 are studied classically and quantum-mechanically (by using Wigner functions W). Difficulties related to the non-positivity of W are bypassed. Equilibrium Wigner functions 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.
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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
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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 StyleÁlvarez-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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