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Physical Review

The Quantum-Mechanical Basis of Statistical Mechanics

Edwin C. Kemble

Published Dec 1, 1939• 11 citations
Publisher

Abstract

Von Neumann's macroscopic entropy definition is not wholly satisfactory because ($a$) it rests on an unsatisfactory characterization of macroscopic variables, ($b$) it fails to make satisfactory connection with the Nernst heat theorem, ($c$) it is based on an unsatisfactory definition of thermal equilibrium, ($d$) it is not applicable to microscopic systems. It is proved that if two systems initially in independent states are placed in thermal contact and isolated, the sum of their microscopic entropies cannot decrease, but will in general increase. This theorem is a substitute for the usual rule regarding the increase in entropy of a single isolated system. Appreciable departures from the canonical state are shown to have a vanishingly small probability in the case of a system subjected to the thermostat experimental procedure described in a preceding paper.

DOI
10.1103/physrev.56.1146

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Von Neumann's macroscopic entropy definition is not wholly satisfactory because (aaa) it rests on an unsatisfactory characterization of macroscopic variables, (bbb) it fails to make satisfactory connection with the Nernst heat theorem, (ccc) it is based on an unsatisfactory definition of thermal equilibrium, (ddd) it is not applicable to microscopic systems. It is proved that if two systems initially in independent states are placed in thermal contact and isolated, the sum of their microscopic entropies cannot decrease, but will in general increase. This theorem is a substitute for the usual rule regarding the increase in entropy of a single isolated system. Appreciable departures from the canonical state are shown to have a vanishingly small probability in the case of a system subjected to the thermostat experimental procedure described in a preceding paper.
To eliminate exceptions to the second law due to fluctuation phenomena, etc., the following changes are necessary. First we must state the laws of thermodynamics in terms of Gibbsian ensembles rather than of individual systems. Second we must sharpen our conception of the experimental procedure required for producing states of thermodynamic equilibrium. It will not suffice to isolate the system and wait. We must have the positive stirring action of thermal contact with a thermostat. Using the new definition of thermal equilibrium we can place statistical mechanics on a more satisfactory footing by showing that the thermostat experimental procedure leads to the canonical ensemble and maximizes v. Neumann's microscopic entropy. (Details of this proof appear in succeeding paper). Difficulties with his macroscopic definition of entropy are avoided by showing that his original microscopic definition is satisfactory when used in conjunction with a proper operational definition of thermodynamic equilibrium. In conclusion the second law of thermodynamics is derived from quantum statistics on the basis of the microscopic definition of entropy.
In this contribution we prove that the entropy of an N-body isolated system can not decrease and the entropy production should be non-negative provided the system possesses an equilibrium state. We define the entropy as a functional of the set of n-particle reduced density operators (n=0,....,N) generalizing the von Neumann fine-grained entropy formula. Additionally, as a consequence of our analysis we find the expression of the equilibrium n-particle reduced density operators which enter the definition of the entropy as well as the dissipated energy in an irreversible process.
The law of increase of entropy is one of the most important laws in science but no exact explanation has been given to the origin of it. Statistical thermodynamics only explains it by using Boltzmann’s statistical interpretation together with not sufficiently verified assumptions. This paper shows that the increase of entropy in an isolated macroscopic system can be reasonably explained with no assumption based on quantum mechanics if we take into account that microscopic particles such as electrons and atomic nuclei not only move in space but also have the basic quality of spontaneously spreading into every possible spatial position. The present work allows us to give a plausible definition to the entropy of an isolated system by using the Hamiltonian operator and a wave function of it and provides a firm basis for thermodynamics. It also gives a new physical meaning of entropy effective for further studies of irreversible processes in nature.
The quantum entropy is usually defined using von Neumann's formula, which measures lack of information and vanishes for pure states. In contrast, we obtain a formula for the entropy of a pure state as it is measured from thermodynamic experiments, solely from the self-entanglement of the wave function, and find strong numerical evidence that the two are in agreement for nonintegrable systems, both for energy eigenstates and for states that are obtained at long times under the evolution of more general initial conditions. This is an extension of Boltzmann's hypothesis for classical systems, relating microscopic motion to thermodynamics.
We treat the problem of the approach to thermal equilibrium by only resorting to quantum dynamics of an isolated macroscopic system. Inspired by the two important works in 2009 and in 1929, we have noted that a condition we call "thermodynamic normality" for a macroscopic observable guarantees the approach to equilibrium (in the sense that a measurement of the observable at time ttt almost certainly yields a result close to the corresponding microcanonical average for a sufficiently long and typical ttt). A crucial point is that we make no assumptions on the initial state of the system, except that its energy is distributed close to a certain macroscopic value. We also present three (rather artificial) models in which the thermodynamic normality can be established, thus providing concrete examples in which the approach to equilibrium is rigorously justified. Note that this kind of results which hold for ANY initial state are never possible in classical systems. We are thus dealing with a mechanism which is peculiar to quantum systems. The present note is written in a self-contained (and hopefully readable) manner. It only requires basic knowledge in quantum physics and equilibrium statistical mechanics.
von Neumann has proved a quantum-mechanical ergodic theorem which deals with a closed macroscopic system and macroscopic observables. Certain unsatisfactory features of this theorem are probably due to the overly-strong idealization of a completely closed system.Since the claims of statistical mechanics go beyond macroscopic systems, there is a need for a more general theorem. The present paper deals with a system of any number of degrees of freedom and with any observable, during interaction with a temperature bath. In order to avoid an assumption of randomness, the bath must be described explicitly and precisely by a time-independent Hamiltonian. The model of the bath is obtained by taking the Gibbs ensemble seriously, i.e., as a set of NNN identical systems interacting through a potential $\ensuremath{\lambda}V$.The following theorem is proved. The time average of the quantum-mechanical expectation value of any observable with respect to any initial state is equal to its statistical average, in the double limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\lambda}\ensuremath{\rightarrow}0$, for the overwhelming majority of all interaction potentials VVV.