The Caldeira–Leggett quantum master equation in Wigner phase space: continued-fraction solution and application to Brownian motion in periodic potentials

García‐Palacios, J. L.; Zueco, David

doi:10.1088/0305-4470/37/45/003

Cited by 45 publications

(62 citation statements)

References 75 publications

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“…, so that the above argument is, due to the actual inclusion of dissipation (γ = 0), fully correct (and entirely convincing, compared to the loose one given previously in Subsection 4.1). Except for the choice made in Equations (47), (48) and (49), the expansion of the non-equilibrium W in terms of standard Hermite polynomials made in Equation (90) is entirely analogous to the one employed in the very extensive work in [35]. The above difficulty that all a γ,n , n = 1, 2, 3, 4, .…”

Section: Conclusion Discussion and Open Problemsmentioning

confidence: 99%

“…The hierarchy in Equations (74), (75) and so on is exact and very general, but it requires to know the eigenfunctions ϕ j (x) and the eigenvalues E j . Our limited aim here was to show that orthogonal polynomials and a non-equilibrium hierarchy exist formally for Equation (35). Then, we shall neither delve further into this nor treat the issue of long-time approximations for Equations (74), (75) and so on.…”

Section: 1mentioning

confidence: 99%

“…A stationary Wigner function is any t-independent solution of Equation (35). We are not interested on arbitrary stationary Wigner functions, but only on a very specific one (denoted by W eq ), namely, that which accounts for the thermal equilibrium state of the qBp, at temperature T = (k B β) −1 , with the hb.…”

Section: Open Quantum-mechanical One-dimensional System Without Dissimentioning

confidence: 99%

“…We shall treat general situations in which the qBp could be out of thermal equilibrium with the hb at t = 0 (and, hence, for t > 0). We shall analyze them by using Equation (35), for the non-equilibrium W . The study will also enable to discuss, as a consistency check, the case in which the qBp be at thermal equilibrium with the hb at any t. The H Q,n (q)'s suggest the following new moments W n (n = 0, 1, 2, .…”

Section: 1mentioning

confidence: 99%

“…For that purpose, we multiply Equation (35) by H Q,n (q/q Q,eq )/q Q,eq , integrate over q and operate, so as to express the resulting equation solely in terms of the W Q,n 's. Again, the issue of performing a check of consistency arises, which will be carried out at a later stage.…”

Section: Acknowledgementsmentioning

confidence: 99%

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Classical and Quantum Models in Non-Equilibrium Statistical Mechanics: Moment Methods and Long-Time Approximations

Alvarez‐Estrada

2012

Entropy

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We consider non-equilibrium open statistical systems, subject to potentials and to external "heat baths" (hb) at thermal equilibrium at temperature T (either with ab initio dissipation or without it). Boltzmann's classical equilibrium distributions generate, as Gaussian weight functions in momenta, orthogonal polynomials in momenta (the position-independent Hermite polynomials H n 's). The moments of non-equilibrium classical distributions, implied by the H n 's, 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.

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Section: Conclusion Discussion and Open Problemsmentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

Section: Open Quantum-mechanical One-dimensional System Without Dissimentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

Section: Acknowledgementsmentioning

confidence: 99%

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