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Fernández ÁlvarezEstrada, Ramón (2014) Nonequilibrium Liouville and Wigner equations: moment methods and longtime approximations. Entropy, 16 (3). pp. 14261461. ISSN 10994300

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Official URL: http://dx.doi.org/10.3390/e16031426
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Abstract
We treat the nonequilibrium evolution of an open oneparticle 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 nonequilibrium threeterm hierarchy for moments fulfills Hermiticity, which allows one to justify an approximate longtime 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 manyparticle systems without hb (by using W_c,eq,), the longtime approximate thermalization for threeterm hierarchies is justified and yields an approximate Lyapunov function and an arrow of time. The largest part of the work treats an open quantum oneparticle system through the nonequilibrium Wigner function, W. W_eq for a repulsive finite square well is reported. W's (< 0 in various cases) are assumed to be quasidefinite functionals regarding their dependences on momentum (q). That yields orthogonal polynomials, H_Q,n (q), for W_eq (and for stationary W_st), nonequilibrium moments, Wn, of W and hierarchies. For the first excited state of the harmonic oscillator, its stationary W_st is a quasidefinite functional, and the orthogonal polynomials and threeterm hierarchy are studied. In general, the nonequilibrium quantum hierarchies (associated with W_eq) for the W_n's are not threeterm ones. As an illustration, we outline a nonequilibrium fourterm hierarchy and its solution in terms of generalized operator continued fractions. Such structures also allow one to formulate longtime approximations, but make it more difficult to justify thermalization. For large thermal and de Broglie wavelengths, the dominant W_eq and a nonequilibrium equation for W are reported: the nonequilibrium hierarchy could plausibly be a threeterm one and possibly not far from Gaussian, and thermalization could possibly be justified.
Item Type:  Article 

Additional Information:  © 2014 by the author. 
Uncontrolled Keywords:  Quantum brownianmotion; Statisticalmechanics; Phasespace; 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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