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Physical Review

Approach to Equilibrium of Quantum Mechanical Gibbsian Ensembles

C. R. Willis

Published Aug 15, 1962• 9 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.127.1405

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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.
In this paper we derive the equations of motion satisfied by the density matrix of an arbitrary physical system in contact with an infinite thermal reservoir of quantum mechanical free particles. We show that for weak but sustained interactions between system and reservoir, the reservoir is represented by a linear time-independent kernel in the system density matrix space. We obtain the kernel for the three possible models of quantum mechanical free particles which are boson and fermion reservoirs in which the system-reservoir interaction conserves the number of particles of the reservoir and a boson reservoir in which reservoir "particles" may be created or destroyed. In order to derive the equations of motion without using the inconsistent "rerandomization of phases" after each interaction, we generalize the Wigner-Weisskopf theory of line broadening of spectral lines to the equations of motion for the density matrix. We next show that the kernel preserves the normalization, positive-definiteness, and Hermiticity of the system density matrix and causes the system to approach equilibrium monotonically. In the limit of a classical reservoir our results include a derivation, without the repeated use of random phasing, of Bloch's equations of motion for the density matrix of a spin system imbedded in a thermal reservoir.
The master equation currently used to describe a two level atom interacting with a single mode field damped by contact with a thermal reservoir (the damped Jaynes-Cummings model) is shown not to have, as its steady state solution, the expected canonical density operator prescribed by the general principles of statistical mechanics for a system in thermal equilibrium. A modified master equation is derived here which satisfies this requirement. Except for a reservoir at zero temperature, this master equation differs from that which is currently used in that the damping terms contain contributions due to the atom-field interaction.
A form of the Schr\"odinger equation is given in which the time variable is replaced by a complex time-temperature variable and the interaction with the heat bath is represented by a stochastic interaction operator with Gaussian white noise. It is shown that at large times the ensemble averaged solution yields the equilibrium spatial probability density of quantum statistical mechanics. As equilibrium is approached, we also show how the master equation for a canonical ensemble can be derived from this form of the Schr\"odinger equation.
Markovian quantum master equations are generally thought to satisfy the following three conditions: (i) they describe the dynamics of time scales that are larger than the reservoir correlation time scale, (ii) these stationary solutions give the thermal equilibrium states with a reservoir, and (iii) they can be written in the Lindblad form. In fact, for single systems such as a single harmonic oscillator and a single two-level system, Markovian quantum master equations satisfying all these conditions can be obtained. However, for composite systems, which consist of more than one subsystem coupled with each other, such equations have not been obtained thus far. In this study, we found that we cannot use the Born–Markov approximation for obtaining quantum master equations for composite systems in the strong-coupling regime, no matter how short the reservoir correlation time is.
We study the structure of the time evolution of the density matrix in contact with a thermal bath in a standard projection operator sheme. The reduced density matrix of the system in the steady state is obtained by tracing out the degree of freedom of the thermal bath from the equilibrium density matrix of the total system. This reduced density matrix is modified by the interaction, and is different from that of the equilibrium of the system alone. We explicitly calculate the contribution of each term in quantum master equation to the realization of the steady state density matrix, and make clear roles of each term. By making use of the role of each term, the properties of the commonly used quantum master equation are examined.