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Generalized quantum Fokker-Planck, diffusion and Smoluchowski equations with true probability distribution functions

Suman Kumar Banik·Bidhan Chandra Bag·Deb Shankar Ray
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Abstract

Traditionally, the quantum Brownian motion is described by Fokker-Planck or diffusion equations in terms of quasi-probability distribution functions, e.g., Wigner functions. These often become singular or negative in the full quantum regime. In this paper a simple approach to non-Markovian theory of quantum Brownian motion using {\it true probability distribution functions} is presented. Based on an initial coherent state representation of the bath oscillators and an equilibrium canonical distribution of the quantum mechanical mean values of their co-ordinates and momenta we derive a generalized quantum Langevin equation in $c$-numbers and show that the latter is amenable to a theoretical analysis in terms of the classical theory of non-Markovian dynamics. The corresponding Fokker-Planck, diffusion and the Smoluchowski equations are the {\it exact} quantum analogues of their classical counterparts. The present work is {\it independent} of path integral techniques. The theory as developed here is a natural extension of its classical version and is valid for arbitrary temperature and friction (Smoluchowski equation being considered in the overdamped limit).

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Suman Kumar Banik, Bidhan Chandra Bag, Deb Shankar Ray. 2002-03-11. Generalized quantum Fokker-Planck, diffusion and Smoluchowski equations with true probability distribution functions. https://doi.org/10.1103/physreve.65.051106

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