{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:32:22Z","timestamp":1760243542040,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2012,2,15]],"date-time":"2012-02-15T00:00:00Z","timestamp":1329264000000},"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 consider non-equilibrium open statistical systems, subject to potentials and to external \u201cheat baths\u201d (hb) at thermal equilibrium at temperature T (either with ab initio dissipation or without it). Boltzmann\u2019s classical equilibrium distributions generate, as Gaussian weight functions in momenta, orthogonal polynomials in momenta (the position-independent Hermite polynomialsHn\u2019s). The moments of non-equilibrium classical distributions, implied by the Hn\u2019s, fulfill a hierarchy: for long times, the lowest moment dominates the evolution towards thermal equilibrium, either with dissipation or without it (but under certain approximation). We revisit that hierarchy, whose solution depends on operator continued fractions. We review our generalization of that moment method to classical closed many-particle interacting systems with neither a hb nor ab initio dissipation: with initial states describing thermal equilibrium at T at large distances but non-equilibrium at finite distances, the moment method yields, approximately, irreversible thermalization of the whole system at T, for long times. Generalizations to non-equilibrium quantum interacting systems meet additional difficulties. Three of them are: (i) equilibrium distributions (represented through Wigner functions) are neither Gaussian in momenta nor known in closed form; (ii) they may depend on dissipation; and (iii) the orthogonal polynomials in momenta generated by them depend also on positions. We generalize the moment method, dealing with (i), (ii) and (iii), to some non-equilibrium one-particle quantum interacting systems. Open problems are discussed briefly.<\/jats:p>","DOI":"10.3390\/e14020291","type":"journal-article","created":{"date-parts":[[2012,2,15]],"date-time":"2012-02-15T12:03:12Z","timestamp":1329307392000},"page":"291-322","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Classical and Quantum Models in Non-Equilibrium Statistical Mechanics: Moment Methods and Long-Time Approximations"],"prefix":"10.3390","volume":"14","author":[{"given":"Ramon F.","family":"Alvarez-Estrada","sequence":"first","affiliation":[{"name":"Departamento de Fisica Teorica I, Facultad de Ciencias Fisicas, Universidad Complutense, 28040 Madrid, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2012,2,15]]},"reference":[{"key":"ref_1","unstructured":"Wallace, D. Reading list for the philosophy of statistical mechanics. Available online: http:\/\/users.ox.ac.uk\/ mert0130\/papers\/smreading.doc."},{"key":"ref_2","unstructured":"Kreuzer, H.J. (1981). Nonequilibrium Thermodynamics and Its Statistical Foundations, Clarendon Press."},{"key":"ref_3","unstructured":"Balescu, R. (1975). Equilibrium and Nonequilibrium Statistical Mechanics, John Wiley and Sons."},{"key":"ref_4","unstructured":"Liboff, R.L. (1998). Kinetic Theory, John Wiley (Interscience). [2nd ed.]."},{"key":"ref_5","unstructured":"Zubarev, D., Morozov, V.G., and R\u00f6pke, G. (1996). Statistical Mechanics of Nonequilibrium Processes, Akademie Verlag."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1103\/PhysRev.40.749","article-title":"On the quantum correction for thermodynamic equilibvrium","volume":"40","author":"Wigner","year":"1932","journal-title":"Phys. Rev."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/0370-1573(84)90160-1","article-title":"Distribution functions in physics: Fundamentals","volume":"106","author":"Hillery","year":"1984","journal-title":"Phys. Rep."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1937","DOI":"10.1088\/0034-4885\/42\/12\/002","article-title":"Foundations of statistical mechanics","volume":"42","author":"Penrose","year":"1979","journal-title":"Rep. Prog. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/S0031-8914(56)80006-2","article-title":"Brownian motion in a field of force and the diffusion theory of chemical reactions","volume":"22","author":"Brinkman","year":"1956","journal-title":"Physica"},{"key":"ref_10","unstructured":"Risken, H. (1989). The Fokker-Planck Equation, Springer. [2nd ed.]."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Coffey, W.T., Kalmykov, Yu. P., and Waldron, J.T. (2004). The Langevin Equation, World Scientific. [2nd ed.].","DOI":"10.1142\/5343"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3361","DOI":"10.1039\/b614554j","article-title":"Wigner function approach to the quantum Bronian motion of a particle in a potential","volume":"9","author":"Coffey","year":"2007","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_13","unstructured":"Abramowitz, M., and Stegun, I.A. (1965). Handbook of Mathematical Functions, Dover."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1002\/andp.20025140502","article-title":"New hierarchy for the Liouville equation, irreversibility and Fokker-Planck-like structures","volume":"11","year":"2002","journal-title":"Ann. Phys. (Leipzig)"},{"key":"ref_15","first-page":"379","article-title":"Liouville and Fokker-Planck dynamics for classical plasmas and radiation","volume":"15","year":"2006","journal-title":"Ann. Phys. (Leipzig)"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"761","DOI":"10.1140\/epja\/i2006-10289-y","article-title":"Nonequilibrium quasi-classical effective meson gas: Thermalization","volume":"31","year":"2007","journal-title":"Eur. Phys. J. A"},{"key":"ref_17","first-page":"391","article-title":"Nonequilibrium quantum anharmonic oscillator and scalar field: High temperature approximations","volume":"18","year":"2009","journal-title":"Ann. Phys. (Berlin)"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1453","DOI":"10.1016\/j.cam.2009.02.061","article-title":"Brownian motion, quantum corrections and a generalization of the Hermite polynomials","volume":"233","year":"2010","journal-title":"J. Comput. Appl. Math."},{"key":"ref_19","first-page":"261","article-title":"Classical systems: Moments, continued fractions, long-time approximations and irreversibility","volume":"1332","year":"2011","journal-title":"AIP Conf. Proc."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/j.cam.2010.01.051","article-title":"Quantum Brownian motion and generalizations of the Hermite polynomials","volume":"236","year":"2011","journal-title":"J. Comput. Appl. Math."},{"key":"ref_21","unstructured":"Abramowitz, M., and Stegun, I.A. (1965). Handbook of Mathematical Functions."},{"key":"ref_22","first-page":"631","article-title":"Is there a \u201ccanonical\u201d non-equilibrium ensemble?","volume":"A447","author":"Penrose","year":"1994","journal-title":"Proc. R. Soc. Lond."},{"key":"ref_23","unstructured":"Louisell, W.H. (1973). Quantum Statistical Properties of Radiation, John Wiley and Sons."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Haken, H. (1970). Laser Theory, Springer. Encyclopedia of Physics, Volume XXV\/2c, Light and Matter Ic.","DOI":"10.1007\/978-3-662-22091-7_1"},{"key":"ref_25","unstructured":"Gardiner, C.W., and Zoller, P. (2004). Quantum Noise, Springer. [3rd ed.]."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Weiss, U. (2008). Quantum Dissipative Systems, World Scientific. [3rd ed.].","DOI":"10.1142\/9789812791795"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Joos, E., Zeh, H.D., Kiefer, C., Giulini, D., Kupsch, J., and Stamatescu, I.-O. (2003). Decoherence and the Appearance of a Classical World in Quantum Theory, Springer. [2nd ed.].","DOI":"10.1007\/978-3-662-05328-7"},{"key":"ref_28","unstructured":"van Kampen, N.G. (2001). Stochastic Processes in Physics and Chemistry, Elsevier."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Breuer, H.-P., and Petruccione, F. (2006). The Theory of Open Quantum Systems, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780199213900.001.0001"},{"key":"ref_30","unstructured":"Haroche, S., and Raimond, J.-M. (2008). Exploring the Quantum, Oxford University Press."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Rivas, A., and Huelga, S.F. (2011). Open Quantum Systems. An Introduction, Springer.","DOI":"10.1007\/978-3-642-23354-8"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"F91","DOI":"10.1088\/1751-8113\/40\/3\/F02","article-title":"Semiclassical Klein-Kramers and Smoluchowski equations for the Brownian motion of a particle in an external potential","volume":"40","author":"Coffey","year":"2007","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1007\/BF01608499","article-title":"On the generators of quantum dynamical semigroups","volume":"48","author":"Lindblad","year":"1976","journal-title":"Commun. Math. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1063\/1.522979","article-title":"Completely positive semigroups of N-level systems","volume":"17","author":"Gorini","year":"1976","journal-title":"J. Math. Phys."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"10735","DOI":"10.1088\/0305-4470\/37\/45\/003","article-title":"The Caldeira-Leggett quantum master equation in Wigner phase space: Continued-fraction solutions and applications to Brownian motion in periodic potentials","volume":"37","author":"Zueco","year":"2004","journal-title":"J. Phys. A Math. 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