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Physical Review

On the Quantum Correction For Thermodynamic Equilibrium

Eugene Paul Wigner

Published Jun 1, 1932• 9,278 citations
Publisher

Abstract

The probability of a configuration is given in classical theory by the Boltzmann formula $\mathrm{exp}[\ensuremath{-}\frac{V}{\mathrm{hT}}]$ where $V$ is the potential energy of this configuration. For high temperatures this of course also holds in quantum theory. For lower temperatures, however, a correction term has to be introduced, which can be developed into a power series of $h$. The formula is developed for this correction by means of a probability function and the result discussed.

DOI
10.1103/physrev.40.749

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The probability of a configuration is given in classical theory by the Boltzmann formula exp[\ensuremath−VhT]\mathrm{exp}[\ensuremath{-}\frac{V}{\mathrm{hT}}]exp[\ensuremath−hTV​] where VVV is the potential energy of this configuration. For high temperatures this of course also holds in quantum theory. For lower temperatures, however, a correction term has to be introduced, which can be developed into a power series of hhh. The formula is developed for this correction by means of a probability function and the result discussed.
Abstract In this paper we explore the following question: can the probabilities constituting the quantum Boltzmann distribution, , be derived from a requirement that the quantum configuration-space distribution for a system in thermal equilibrium be very similar to the corresponding classical distribution? It is certainly to be expected that the quantum distribution in configuration space will approach the classical distribution as the temperature approaches infinity, and a well-known equation derived from the Boltzmann distribution shows that this is generically the case. Here we ask whether one can reason in the opposite direction, that is, from quantum-classical agreement to the Boltzmann probabilities. For two of the simple examples we consider—a particle in a 1D box and a simple harmonic oscillator—this approach leads to probability distributions that provably approach the Boltzmann probabilities at high temperature, in the sense that the Kullback–Leibler divergence between the distributions approaches zero.
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A recently proposed Boltzmann local equilibrium Wigner function for massive spin-1/2 particles is generalized to the case of Fermi-Dirac statistics. The resulting formula ensures the correct normalization of the mean polarization vector and reproduces the generalized thermodynamic relations with spin that were obtained in earlier studies. Moreover, we show that the macroscopic currents constructed from the Fermi-Dirac Wigner function can be obtained as derivatives of a suitably defined generating function with respect to the Lagrange multipliers (temperature, hydrodynamic flow, and chemical potentials). The identified generating function also indicates that the underlying framework can be classified as a divergence-type theory.
The function F{N} which gives the relative probability of a configuration {N} is studied for the case of a quantum-mechanical system to which classical statistics can be applied. By means of a device originally used in quantum electrodynamics F{N} is obtained in a new form which is very closely related to the form of the analogous classical distribution FCL{N}. The new form of F{N} shows clearly how the difference between F{N} and FCL{N} is related to the uncertainty principle. As an illustration of the utility of the methods presented here several applications are presented. The first application is a high temperature development of F{N} as a power series in ℏ/kT which is carried through to the fourth order. The second application is the development of F{N} as a series of configuration space integrals. A previously published low temperature development of F{N} is obtained in a simple manner, and it is shown that all higher terms in the development can be written down explicitly by starting from the new form of F{N}. Finally, several integral equations for F{N} are presented which give physical insight into its structure, by establishing mathematical analogies with such physical processes as neutron diffusion.
A finite-temperature many-body perturbation theory is presented, which expands in power series the electronic grand potential, chemical potential, internal energy, and entropy on an equal footing. Sum-over-states and sum-over-orbitals analytical formulas for the second-order perturbation corrections to these thermodynamic properties are obtained in a time-independent, nondiagrammatic, algebraic derivation, relying on the sum rules of the Hirschfelder–Certain degenerate perturbation energies in a degenerate subspace as well as nine algebraic identities for the zeroth-order thermal averages of one- through four-indexed quantities and products thereof. They reproduce numerically exactly the benchmark data obtained as the numerical derivatives of the thermal-full-configuration-interaction results for a wide range of temperatures.