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Entropy 2014, 16(3), 1426-1461; doi:10.3390/e16031426
Article

Non-Equilibrium Liouville and Wigner Equations: Moment Methods and Long-Time Approximations

Departamento de Fisica Teorica I, Facultad de Ciencias Fisicas, Universidad Complutense, Madrid 28040, Spain
Received: 12 December 2013 / Revised: 7 February 2014 / Accepted: 24 February 2014 / Published: 11 March 2014
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Abstract

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, 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’s 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’s (< 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’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 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.
Keywords: Liouville and Wigner distributions and non-equilibrium equations; equilibrium solutions and orthogonal polynomials; non-equilibrium moments; long-time approximations; low temperature regime Liouville and Wigner distributions and non-equilibrium equations; equilibrium solutions and orthogonal polynomials; non-equilibrium moments; long-time approximations; low temperature regime
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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Álvarez-Estrada, R.F. Non-Equilibrium Liouville and Wigner Equations: Moment Methods and Long-Time Approximations. Entropy 2014, 16, 1426-1461.

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