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arXiv:2006.01121 · Journal of Computational and Theoretical Transport

Adding Decoherence to the Wigner Equation

Luigi Barletti, Giovanni Frosali, Elisa Giovannini

Published Apr 16, 2018• 3 citations
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Abstract

Starting from the detailed description of the single-collision decoherence mechanism proposed by Adami, Hauray and Negulescu (2016 R. Adami, M. Hauray, C. Negulescu. 2016. Decoherence for a heavy particle interacting with a light particle: new analysis and numerics. Comm. Math. Sci. 14 (5):1373–1415.[Crossref], [Web of Science ®] , [Google Scholar]), we derive a Wigner equation endowed with a decoherence term of a fairly general form. This equation is shown to contain well known decoherence models, such as the Wigner–Fokker–Planck equation, as particular cases. The effect of the decoherence mechanism on the dynamics of the macroscopic moments (density, current, energy) is illustrated by deriving the corresponding set of balance laws. The issue of large-time asymptotics of our model is addressed in the particular, although physically relevant, case of Gaussian solutions. It is shown that the addition of a Caldeira–Legget friction term provides the asymptotic behavior that one expects on the basis of physical considerations.

DOI
10.1080/23324309.2018.1520732

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Starting from the detailed description of the single-collision decoherence mechanism proposed by Adami, Hauray and Negulescu (2016 R. Adami, M. Hauray, C. Negulescu. 2016. Decoherence for a heavy particle interacting with a light particle: new analysis and numerics. Comm. Math. Sci. 14 (5):1373–1415.[Crossref], [Web of Science ®] , [Google Scholar]), we derive a Wigner equation endowed with a decoherence term of a fairly general form. This equation is shown to contain well known decoherence models, such as the Wigner–Fokker–Planck equation, as particular cases. The effect of the decoherence mechanism on the dynamics of the macroscopic moments (density, current, energy) is illustrated by deriving the corresponding set of balance laws. The issue of large-time asymptotics of our model is addressed in the particular, although physically relevant, case of Gaussian solutions. It is shown that the addition of a Caldeira–Legget friction term provides the asymptotic behavior that one expects on the basis of physical considerations.
Dopo un’introduzione sui fondamenti matematici della Meccanica Quantistica e la sua formulazione wigneriana, si ricava un’equazione 1-dimensionale di Wigner con termine di decoerenza partendo dalla descrizione dettagliata di R. Adami, M. Hauray, C. Negulescu per la decoerenza di una particella pesante che interagisce con una particella leggera. Si mostra che il modello ottenuto ne contiene, quali casi particolari, altri già utilizzati per descrivere il fenomeno della decoerenza quantistica, come ad esempio l’equazione di Wigner–Fokker–Planck o le funzioni di Wigner con lunghezza di coerenza finita. Si indaga l’effetto della decoerenza sulla dinamica delle quantità macroscopiche (densità, corrente, energia) attraverso le corrispondenti leggi di bilancio. Si applica poi il modello ricavato ad una situazione di interesse fisico mediante simulazioni numeriche: un processo di tunneling in ambiente decoerente. Si investiga la questione asintotica per tempi lunghi nel caso di soluzioni gaussiane e si accenna al caso generale spazio omogeneo. Si mostra che l’aggiunta di un termine di frizione quantistica di tipo Caldeira-Legget all’equazione di Wigner con decoerenza è capace di realizzare il comportamento asintotico prevedibile sulla base di considerazioni di natura fisica.
This paper studies the long-time behavior of a spatially homogeneous Wigner equation describing a quantum particle subject to decoherence. The model extends the classical Wigner–Fokker–Planck equation by including a general decoherence term derived from microscopic collision mechanisms. An explicit representation of the equilibrium state is obtained, and a necessary and sufficient condition ensuring that the equilibrium state belongs to L 2 is established, highlighting the balance between friction and decoherence. Under suitable assumptions on the initial data, exponential convergence in L 2 toward equilibrium is rigorously proved.
This work provides a detailed derivation of a generalized quantum Fokker-Planck equation (GQFPE) appropriate for photo-induced quantum dynamical processes. The path integral method pioneered by Caldeira and Leggett (CL) [Physica A 121, 587 (1983)] is extended by utilizing a nonequilibrium influence functional applicable to different baths for the ground and the excited electronic states. Both nonequilibrium and non-Markovian effects are accounted for consistently by expanding the paths in the exponents of the influence functional up to the second order with respect to time. This procedure results in approximations involving only single time integrations for the exponents of the influence functional but with additional time dependent boundary terms that have been ignored in previous works. The boundary terms complicate the derivation of a time evolution equation but do not affect position dependent physical observables or the dynamics in the steady state limit. For an effective density operator with the boundary terms factored out, a time evolution equation is derived, through short time expansion of the effective action and Gaussian integration in analytically continued complex domain of space. This leads to a compact form of the GQFPE with time dependent kernels and additional terms, which renders the resulting equation to be in the Dekker form [Phys. Rep. 80, 1 (1981)]. Major terms of the equation are analyzed for the case of Ohmic spectral density with Drude cutoff, which shows that the new GQFPE satisfies the positive definiteness condition in medium to high temperature limit. Steady state limit of the GQFPE is shown to approach the well-known expression derived by CL in the high temperature and Markovian bath limit and also provides additional corrections due to quantum and non-Markovian effects of the bath.
Starting from the microscopic description of a normal fluid in terms of any kind of local interacting many-particle theory we present a well defined step by step procedure to derive the hydrodynamic equations for the macroscopic phenomena. We specify the densities of the conserved quantities as the relevant hydrodynamic variables and apply the methods of non-equilibrium statistical mechanics with projection operator techniques. As a result we obtain time-evolution equations for the hydrodynamic variables with three kinds of terms on the right-hand sides: reversible, dissipative and fluctuating terms. In their original form these equations are completely exact and contain nonlocal terms in space and time which describe nonlocal memory effects. Applying a few approximations the nonlocal properties and the memory effects are removed. As a result we find the well known hydrodynamic equations of a normal fluid with Gaussian fluctuating forces. In the following we investigate if and how the time-inversion invariance is broken and how the second law of thermodynamics comes about. Furthermore, we show that the hydrodynamic equations with fluctuating forces are equivalent to stochastic Langevin equations and the related Fokker-Planck equation. Finally, we investigate the fluctuation theorem and find a modification by an additional term.