{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:27:00Z","timestamp":1760243220083,"version":"build-2065373602"},"reference-count":50,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2014,3,11]],"date-time":"2014-03-11T00:00:00Z","timestamp":1394496000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We treat the non-equilibrium evolution of an open one-particle statistical system, subject to a potential and to an external \u201cheat bath\u201d (hb) with negligible dissipation. For the classical equilibrium Boltzmann distribution, Wc,eq, a non-equilibrium three-term hierarchy for moments fulfills Hermiticity, which allows one to justify an approximate long-time thermalization. That gives partial dynamical support to Boltzmann\u2019s Wc,eq, out of the set of classical stationary distributions, Wc;st, also investigated here, for which neither Hermiticity nor that thermalization hold, in general. For closed classical many-particle systems without hb (by using Wc,eq), the long-time approximate thermalization for three-term hierarchies is justified and yields an approximate Lyapunov function and an arrow of time. The largest part of the work treats an open quantum one-particle system through the non-equilibrium Wigner function, W. Weq for a repulsive finite square well is reported. W\u2019s (&lt; 0 in various cases) are assumed to be quasi-definite functionals regarding their dependences on momentum (q). That yields orthogonal polynomials, HQ,n(q), for Weq (and for stationary Wst), non-equilibrium moments, Wn, of W and hierarchies. For the first excited state of the harmonic oscillator, its stationary Wst is a quasi-definite functional, and the orthogonal polynomials and three-term hierarchy are studied. In general, the non-equilibrium quantum hierarchies (associated with Weq) for the Wn\u2019s are not three-term ones. As an illustration, we outline a non-equilibrium four-term hierarchy and its solution in terms of generalized operator continued fractions. Such structures also allow one to formulate long-time approximations, but make it more difficult to justify thermalization. For large thermal and de Broglie wavelengths, the dominant Weq and a non-equilibrium equation for W are reported: the non-equilibrium hierarchy could plausibly be a three-term one and possibly not far from Gaussian, and thermalization could possibly be justified.<\/jats:p>","DOI":"10.3390\/e16031426","type":"journal-article","created":{"date-parts":[[2014,3,11]],"date-time":"2014-03-11T12:09:38Z","timestamp":1394539778000},"page":"1426-1461","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Non-Equilibrium Liouville and Wigner Equations: Moment Methods and Long-Time Approximations"],"prefix":"10.3390","volume":"16","author":[{"given":"Ramon","family":"\u00c1lvarez-Estrada","sequence":"first","affiliation":[{"name":"Departamento de Fisica Teorica I, Facultad de Ciencias Fisicas, Universidad Complutense, Madrid 28040, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2014,3,11]]},"reference":[{"key":"ref_1","unstructured":"Wallace, D. 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