White Noise Approach to the Low Density Limit

Abstract

The white noise approach to the investigation of the dynamics of a quantum particle interacting with a dilute and in general non-equilibrium gaseous environment in the low density limit is outlined. The low density limit is the kinetic Markovian regime when only pair collisions (i.e., collisions of the test particle with one particle of the gas at one time moment) contribute to the dynamics. In the white noise approach one first proves that the appropriate operators describing the gas converge in the sense of appropriate matrix elements to certain operators of quantum white noise. Then these white noise operators are used to derive quantum white noise and quantum stochastic equations describing the approximate dynamics of the total system consisting of the particle and the gas. The derivation is given ab initio, starting from the exact microscopic quantum dynamics. The limiting dynamics is described by a quantum stochastic equation driven by a quantum Poisson process. This equation then applied to the derivation of quantum Langevin equation and linear Boltzmann equation for the reduced density matrix of the test particle. The first part of the paper describes the approach which was developed by L. Accardi, I.V. Volovich and the author and uses the Fock-antiFock (or GNS) representation for the CCR algebra of the gas. The second part presents the approach to the derivation of the limiting equations directly in terms of the correlation functions, without use of the Fock-antiFock representation. This approach simplifies the derivation and allows to express the strength of the quantum number process directly in terms of the one-particle S-matrix.

Quantum linear Boltzmann equation

“…Finally, we note that a large number of works can be found in the literature which are related to the present subject. While it is impossible to do justice to all authors, we mention a few recent articles related to the derivation of a quantum kinetic equation in the spirit of the Boltzmann equation, both linear [70,71,72,73,74,75,76,77,78,79,80] and nonlinear [11,12,13,14,15,16,17,20,81,82,83,23,84,85].…”

Normalization of collisional decoherence: squaring the delta function, and an independent cross-check

“…I also wish to thank Klaus Hornberger for an informative email correspondence, and for noting a recent paper [13], in which he extends the method of ref [4] to the case of a finite mass Brownian particle. Finally, Alexander Pechen has pointed out a recent paper [14], giving further references, pertaining to the microscopic derivation of the master equation for a quantum system interacting with a dilute environment.…”