Non-equilibrium Liouville and Wigner equations: moment methods and long-time approximations

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We treat the non-equilibrium evolution of an open one-particle statistical system, subject to a potential and to an external "heat bath" (hb) with negligible dissipation. For the classical equilibrium Boltzmann distribution, W_c,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's W_c,eq, out of the set of classical stationary distributions, W_c,st, also investigated here, for which neither Hermiticity nor that thermalization hold, in general. For closed classical many-particle systems without hb (by using W_c,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. W_eq for a repulsive finite square well is reported. W's (< 0 in various cases) are assumed to be quasi-definite functionals regarding their dependences on momentum (q). That yields orthogonal polynomials, H_Q,n (q), for W_eq (and for stationary W_st), non-equilibrium moments, W-n, of W and hierarchies. For the first excited state of the harmonic oscillator, its stationary W_st is a quasi-definite functional, and the orthogonal polynomials and three-term hierarchy are studied. In general, the non-equilibrium quantum hierarchies (associated with W_eq) for the W_n's 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 W_eq 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.
© 2014 by the author. ©2014 Multidisciplinary Digital Publishing Institute. The author is grateful to the Editorial Office of Entropy and to the Guest Editor, Gian Paolo Beretta, for inviting him to contribute to the Special Issue Advances in Methods and Foundations on Non-Equilibrium Thermodynamics. The author acknowledges the financial support of Project FIS2012-35719-C02-01, Ministerio de Economia y Competitividad, Spain. He is an associate member of BIFI (Instituto de Biocomputacion y Fisica de los Sistemas Complejos), Universidad de Zaragoza, Zaragoza, Spain. Several discussions with Gabriel F. Calvo on the dynamics based either upon random walkers or on average densities have been quite useful.
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