{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:33:17Z","timestamp":1760239997004,"version":"build-2065373602"},"reference-count":56,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2019,2,14]],"date-time":"2019-02-14T00:00:00Z","timestamp":1550102400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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 \u201cheat bath\u201d (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\u2019s Gaussian classical equilibrium distribution     W  c , e q, as weight function to generate orthogonal polynomials (H n\u2019s) in momenta. The moments of     W c, implied by the     H n\u2019s, 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.<\/jats:p>","DOI":"10.3390\/e21020179","type":"journal-article","created":{"date-parts":[[2019,2,14]],"date-time":"2019-02-14T11:54:13Z","timestamp":1550145253000},"page":"179","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Non-Equilibrium Liouville and Wigner Equations: Classical Statistical Mechanics and Chemical Reactions for Long Times"],"prefix":"10.3390","volume":"21","author":[{"given":"Ramon F.","family":"\u00c1lvarez-Estrada","sequence":"first","affiliation":[{"name":"Departamento de Fisica Teorica, Facultad de Ciencias Fisicas, Universidad Complutense, 28040 Madrid, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2019,2,14]]},"reference":[{"key":"ref_1","unstructured":"Wallace, D. (2019, February 14). Reading List for Advanced Philosophy of Physics: The Philosophy of Statistical Mechanics. 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