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THEORY OF STOCHASTIC PROCESSES IN

N. Barabanenkov

Published Jan 1, 1980• 0 citations
OpenAlex

Abstract

The stochastic process of the motion of a test particle in a dynamical system with short~ range two-body interaction potential is considered. The method of stochastic perturbation theory in the low density and the procedure of renormalization of the Green's function of the test particle and the formalism of collision operators for the Liouville equation are used to give a quantum generalization of Bogolyubov's kinetic equation [I]+ this leading to a nonexponential law of decrease of the correlation function of the velocity of the test particle in a system of classical hard spheres over long time intervals

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The stochastic process of the motion of a test particle in a dynamical system with short~ range two-body interaction potential is considered. The method of stochastic perturbation theory in the low density and the procedure of renormalization of the Green's function of the test particle and the formalism of collision operators for the Liouville equation are used to give a quantum generalization of Bogolyubov's kinetic equation [I]+ this leading to a nonexponential law of decrease of the correlation function of the velocity of the test particle in a system of classical hard spheres over long time intervals
Dynamics of quantum systems which are stochastically perturbed by linear coupling to the reservoir can be studied in terms of quantum stochastic differential equations (for example, quantum stochastic Liouville equation and quantum Langevin equation).In order to work it out one needs to define the quantum Brownian motion.As far as only its boson version has been known until recently, in the present paper we present the definition which makes it possible to consider the fermion Brownian motion as well.
Within the framework of the first-order smoothing approximation and the long-time, Markovian approximation, kinetic equations are derived for the stochastic Wigner equation (the exact equation of evolution of the phase-space Wigner distribution function) and the stochastic Liouville equation (correspondence limit approximation) associated with the quantized motion of a particle described by a stochastic Schrödinger equation. In the limit of weak fluctuations and long times, the transport equation for the average probability density of the particle in momentum space which was reported recently by Papanicolaou is recovered. Also, on the basis of the Novikov functional formalism, it is established that several of the approximate kinetic equations derived in this paper are identical to the exact statistical equations in the special case that the potential field is a δ-correlated (in time), homogeneous, wide-sense stationary, Gaussian process.
When a dynamical system has a perturbation which is considered as a stochastic process, the Liouville equation for the system in the phase space or the space of quantum-mechanical density operators is a sort of stochastic equation. The ensemble average of its formal integral defines the relaxation operator Φ(t) of the system. By the definition Φ(t) = exp K(t), the cumulant function K(t) may be introduced. Some general properties are first discussed for a simple example of an oscillator with random frequency modulation and then, concepts of slow and fast modulation are considered. These concepts can be generalized to more general types of stochastic Liouville equations. It is shown that by various possibilities of defining generalized exponential functions, this approach may be useful to understand some essential features of the problem from an unified point of view.
The problem of many-particle correlations in different approaches to the quantum kinetic theory is treated on the basis of Zubarev's method of the nonequilibrium statistical operator.It is shown that long-lived correlations can be incorporated through boundary conditions for reduced manyparticle density matrices and the nonequilibrium real-time Green functions.Within the method of Green functions the boundary conditions are conveniently formulated in terms of the "mixed" Green functions defined on a directed contour with the real-time evolution governed by the Hamiltonian of the system and the "imaginary-time" evolution governed by the entropy operator.The perturbation expansion of the mixed Green function is obtained in terms of the interaction part of the Hamiltonian and the correlation part of the entropy operator.
A rigorous derivation of quantum Langevin equation from microscopic dynamics in the low density limit is given. We consider a quantum model of a microscopic system (test particle) coupled with a reservoir (gas of light Bose particles) via interaction of scattering type. We formulate a mathematical procedure (the so-called stochastic golden rule) which allows us to determine the quantum Langevin equation in the limit of large time and small density of particles of the reservoir. The quantum Langevin equation describes not only dynamics of the system but also the reservoir. We show that the generator of the corresponding master equation has the Lindblad form of most general generators of completely positive semigroups.
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 SSS-matrix.