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Fernández Álvarez-Estrada, Ramón
(2014)
*Non-equilibrium Liouville and Wigner equations: moment methods and long-time approximations.*
Entropy, 16
(3).
pp. 1426-1461.
ISSN 1099-4300

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Official URL: http://dx.doi.org/10.3390/e16031426

## 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, 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.

Item Type: | Article |
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Additional Information: | © 2014 by the author. |

Uncontrolled Keywords: | Quantum brownian-motion; Statistical-mechanics; Phase-space; Dynamics; Irreversibility; Oscillator; Operators; Lionville; Systems; Field |

Subjects: | Sciences > Physics |

ID Code: | 34712 |

Deposited On: | 09 Dec 2015 17:47 |

Last Modified: | 09 Dec 2015 17:47 |

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