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Journal of Physics C Solid State Physics

Quantum Fokker-Planck equation

L -D Chang, David Waxman

Published Nov 10, 1985• 22 citations
Publisher

Abstract

The dynamics of quantum systems subject to dissipation is a subject of fundamental importance which has recently been of great interest due to its relation to macroscopic quantum phenomena, In particular the behaviour of the magnetic flux trapped in a SQUID is an example of this. The authors derive the quantum mechanical analogue of the Fokker-Planck equation which describes such systems. The major difference between the quantum and classical Fokker-Planck equations arises from the presence of a quantum mechanical memory term.

DOI
10.1088/0022-3719/18/31/019

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The dynamics of quantum systems subject to dissipation is a subject of fundamental importance which has recently been of great interest due to its relation to macroscopic quantum phenomena, In particular the behaviour of the magnetic flux trapped in a SQUID is an example of this. The authors derive the quantum mechanical analogue of the Fokker-Planck equation which describes such systems. The major difference between the quantum and classical Fokker-Planck equations arises from the presence of a quantum mechanical memory term.
This paper is intended to provide the reader with a review of the authors’ latest results dealing with the modeling of quantum dissipation/diffusion effects at the level of Schrodinger systems, in connection with the corresponding phase space and fluid formulations of such kind of phenomena, especially in what concerns the role of the Fokker–Planck mechanism in the description of open quantum systems and the macroscopic dynamics associated with some viscous hydrodynamic models of Euler and Navier–Stokes type.
A study is provided of dynamics and dissipative tunneling in a symmetric quartic double well potential. The numerical solution for the position autocorrelation function obtained through the Wigner–Fokker–Planck equation is compared with numerically exact results of Stockburger and Mak [J. Chem. Phys. 110, 4983 (1999)]. We find that the Wigner–Fokker–Planck dynamics agree well with the numerically exact computations, they account for both quantum coherences as well as quantum tunneling phenomena. This, in contrast to the mixed quantum classical approximation, which does not perform as well.
We develop a generalized quantum Fokker-Planck theory in a non-Gaussian-Markovian model bath. The semiclassical bath adopted in this work is charactered by three parameters. One denotes the strength of system-bath coupling and the other two are chosen to interpolate smoothly the solvation dynamics between the long- and short-time regimes. The fluctuation-dissipation relation in this model bath is analyzed in detail. Based on this model bath, we derive two sets of coupled Fokker-Planck equations. These two equation sets are equivalent in the second order of system-bath coupling but different in the higher orders. The corresponding reduced Liouville equation in one set of the Fokker-Planck formulation is characterized by a memory relaxation kernel, while that in the other is by a local-time relaxation tensor. Each resulting set of Fokker-Planck equations involves only the reduced density operator and a series of well-characterized Hilbert-space relaxation operators. The present theory is valid for arbitrary time-dependent Hamiltonians and is applicable to the study of quantum coherence and relaxation in various dynamic systems.
A study of the non-dissipative Brownian motion in vacuum is presented. The noise source associated to the stochastic process assumed in this work is vacuum fluctuations of some quantum field capable of interact with a massive particle. For the associated Fokker-Planck equation we do a perturbation theory which has terms presenting dynamics similar to that satisfied by the probability amplitude in Quantum Theory even though the physical interpretation be classical. In particular classical observables present expected values that coincide with those calculated using Quantum Mechanics.
A nonlinear Schrodinger equation describing how a quantum particle interacts with its surrounding reservoir is derived from the Wigner–Fokker–Planck equation (WFPE) via stochastic (Nelsonian) mechanical techniques. This model can be reduced just to a logarithmic Schrodinger equation through a suitable gauge transformation that allows one to explore its steady state dynamics and makes its mathematical and numerical analysis more tractable. The transient behavior of the standard deviation from the mean position associated with its solutions is also studied numerically and compared with that stemming from the WFPE.
Relaxation according to Fokker-Planck equations is a standard scenario in classical statistical mechanics. It is, however, not obvious how such a Fokker-Planck dynamics may emerge within a closed, finite quantum system. We present an analysis of a system comprising sixteen spins in which spatial inhomogeneities of the magnetization relax approximately in accord with a standard Fokker-Planck equation for a Brownian particle in a parabolic potential. While a single, specfic example is addressed in detail, arguments which indicate that the above findings may hold for a wider range of systems are given.