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Journal of Mathematical Physics

Physical applications of multiplicative stochastic processes

Ronald F. Fox

Published Jan 1, 1973• 14 citations
Publisher

Abstract

The theory of multiplicative stochastic processes has been shown to lead to a density matrix description of nonequilibrium quantum mechanical phenomena. In the present paper a detailed treatment of the approach to the uniform, microcanonical, and canonical equilibrium density matrices is presented. The canonical equilibrium density matrix is approached by the density matrix which represents a subsystem in contact with a constant temperature heat reservoir.

DOI
10.1063/1.1666166

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The theory of multiplicative stochastic processes has been shown to lead to a density matrix description of nonequilibrium quantum mechanical phenomena. In the present paper a detailed treatment of the approach to the uniform, microcanonical, and canonical equilibrium density matrices is presented. The canonical equilibrium density matrix is approached by the density matrix which represents a subsystem in contact with a constant temperature heat reservoir.
The multiplicative stochastic process treatment of the time development of the density matrix for a subsystem in contact with a heat reservoir is applied to the specific problem of the relaxation of a nuclear magnetic moment which is interacting with a fluctuating magnetic environment. A model for the fluctuating interaction Hamiltonian, appropriate for the magnetic moment case, is presented, and the Bloch equations for nuclear magnetic relaxation are constructed as a consequence. Agreement with empirical observations is noted.
It is argued that the expression − KB Trace [〈ρ (t) 〉 ln〈ρ (t) 〉], which appears in a stochastic treatment of the dynamics of the density matrix is indeed the nonequilibrium entropy. The reasoning involves consideration of the time evolution of the free energy for a relaxing magnetic moment in a fluctuating magnetic environment. It is shown that the H theorem, and the monotonic decrease of the free energy, as described by Pauli’s master equation can be generalized to the full density matrix, at least for the case of magnetic relaxation, which requires the presence of off-diagonal density matrix elements.
In this paper we study the approach to equilibrium of an arbitrary physical system in contact with a thermal reservoir when the internal dynamics of the system is described by its Hamiltonian and the interaction with the reservoir can be described by a linear stochastic kernel which in general can be time dependent. We find a generalization of the Markov process represented by the master equation to the space of density matrices and we prove that the necessary and sufficient condition that any solution of an equation of our general structure approach equilibrium monotonically is that equilibrium is a stationary solution. We prove that the Helmholtz free energy is a minimum as a functional of the density matrix and thus is a suitable measure of deviation from canonicity. We also find another measure of deviation which is equally good and easier to use. We show, as an example, that Bloch's microscopic density matrices fulfill our criteria and thus approach equilibrium monotonically. Finally, we extend our results for the thermal reservoir to more general reservoirs.
Descriptions of molecular systems usually refer to two distinct theoretical frameworks. On the one hand the quantum pure state, i.e., the wavefunction, of an isolated system is determined to calculate molecular properties and their time evolution according to the unitary Schrödinger equation. On the other hand a mixed state, i.e., a statistical density matrix, is the standard formalism to account for thermal equilibrium, as postulated in the microcanonical quantum statistics. In the present paper an alternative treatment relying on a statistical analysis of the possible wavefunctions of an isolated system is presented. In analogy with the classical ergodic theory, the time evolution of the wavefunction determines the probability distribution in the phase space pertaining to an isolated system. However, this alone cannot account for a well defined thermodynamical description of the system in the macroscopic limit, unless a suitable probability distribution for the quantum constants of motion is introduced. We present a workable formalism assuring the emergence of typical values of thermodynamic functions, such as the internal energy and the entropy, in the large size limit of the system. This allows the identification of macroscopic properties independently of the specific realization of the quantum state. A description of material systems in agreement with equilibrium thermodynamics is then derived without constraints on the physical constituents and interactions of the system. Furthermore, the canonical statistics is recovered in all generality for the reduced density matrix of a subsystem.
A new approach to quantum Markov processes is developed and the corresponding Fokker-Planck-like equation is derived. The latter was examined to reproduce known results from classical and quantum physics. It was also applied to the phase-space description of a mechanical system thus leading to a new treatment of this problem different from the Wigner presentation. The equilibrium probability density obtained in the mixed coordinate-momentum space is a reasonable extension of the Gibbs canonical distribution.