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Physical Review

Distribution Functions and Quantum Statistics

R. U. Ayres

Published Dec 1, 1960• 1 citation
Publisher

Abstract

A formulation of quantum statistical mechanics is given, in terms of distribution functions. It is shown that all quantities of interest are obtained directly from nonlocal distribution functions ${\ensuremath{\gamma}}^{(1)}({{x}_{1}}^{\ensuremath{'}}, {x}_{1}, \ensuremath{\beta}), {\ensuremath{\gamma}}^{(2)}({{x}_{1}}^{\ensuremath{'}}, {{x}_{2}}^{\ensuremath{'}}, {x}_{1}{x}_{2}, \ensuremath{\beta}), \ensuremath{\cdots}.$ In a uniform system the Fourier transform of ${\ensuremath{\gamma}}^{(1)}({{x}_{1}}^{\ensuremath{'}}, {x}_{1}, \ensuremath{\beta})$ is the distribution of particles in $k$ space, $n(k, \ensuremath{\beta})$. Various schemes for calculating these distribution functions directly are discussed. An approximate integral equation for $n(k, \ensuremath{\beta})$ is found, which can be solved by a converging iteration process. Some remarks are included on the application of the virial theorem.

DOI
10.1103/physrev.120.1557

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A formulation of quantum statistical mechanics is given, in terms of distribution functions. It is shown that all quantities of interest are obtained directly from nonlocal distribution functions \ensuremathγ(1)(x1\ensuremath′,x1,\ensuremathβ),\ensuremathγ(2)(x1\ensuremath′,x2\ensuremath′,x1x2,\ensuremathβ),\ensuremath⋯.{\ensuremath{\gamma}}^{(1)}({{x}_{1}}^{\ensuremath{'}}, {x}_{1}, \ensuremath{\beta}), {\ensuremath{\gamma}}^{(2)}({{x}_{1}}^{\ensuremath{'}}, {{x}_{2}}^{\ensuremath{'}}, {x}_{1}{x}_{2}, \ensuremath{\beta}), \ensuremath{\cdots}.\ensuremathγ(1)(x1​\ensuremath′,x1​,\ensuremathβ),\ensuremathγ(2)(x1​\ensuremath′,x2​\ensuremath′,x1​x2​,\ensuremathβ),\ensuremath⋯. In a uniform system the Fourier transform of \ensuremathγ(1)(x1\ensuremath′,x1,\ensuremathβ){\ensuremath{\gamma}}^{(1)}({{x}_{1}}^{\ensuremath{'}}, {x}_{1}, \ensuremath{\beta})\ensuremathγ(1)(x1​\ensuremath′,x1​,\ensuremathβ) is the distribution of particles in kkk space, n(k,\ensuremathβ)n(k, \ensuremath{\beta})n(k,\ensuremathβ). Various schemes for calculating these distribution functions directly are discussed. An approximate integral equation for n(k,\ensuremathβ)n(k, \ensuremath{\beta})n(k,\ensuremathβ) is found, which can be solved by a converging iteration process. Some remarks are included on the application of the virial theorem.
An attempt is made to interpret quantum mechanics as a statistical theory, or more exactly as a form of non-deterministic statistical dynamics. The paper falls into three parts. In the first, the distribution functions of the complete set of dynamical variables specifying a mechanical system (phase-space distributions), which are fundamental in any form of statistical dynamics, are expressed in terms of the wave vectors of quantum theory. This is shown to be equivalent to specifying a theory of functions of non-commuting operators, and may hence be considered as an interpretation of quantum kinematics . In the second part, the laws governing the transformation with time of these phase-space distributions are derived from the equations of motion of quantum dynamics and found to be of the required form for a dynamical stochastic process. It is shown that these phase-space transformation equations can be used as an alternative to the Schrödinger equation in the solution of quantum mechanical problems, such as the evolution with time of wave packets, collision problems and the calculation of transition probabilities in perturbed systems; an approximation method is derived for this purpose. The third part, quantum statistics , deals with the phase-space distribution of members of large assemblies, with a view to applications of quantum mechanics to kinetic theories of matter. Finally, the limitations of the theory, its uniqueness and the possibilities of experimental verification are discussed.
In this paper, four distinct ideas are combined, which under a wide range of circumstances can give very rapidly converging series expansions for functional integrals. (1) Expansion of the functional being integrated in functional Taylor series. In the familiar case arising in quantum statistical mechanics, that of the Wiener integral of exp[−∫V(x(t))dt],V(x) being the perturbing potential, this is equivalent to expanding the characteristic functional of the probability functional of V(x(t)) in central moments of V(x(t)). The lowest-order term of the series is the approximation obtained by Feynman and Hibbs through a variational method. (2) Transfer of the harmonic term of the potential, when the functional integral is the quantum-statistical density matrix (Green's function of the Bloch equation), to the weighting function. This transforms the functional integral from a Wiener to an Uhlenbeck-Ornstein integral. The formal expressions for the terms of the expansion are somewhat more complicated, but they can be worked out, and the result is a great improvement in the speed of convergence of the series with decreasing temperature and/or decreasing relative magnitude of the anharmonic part of the potential. (3) ``Reservation of variables'' in the integration. This amounts to breaking the averaging process down into an average over subsets of the distributed random function (conditional average), followed by an averaging of these averages. Any step of this kind (it may be repeated within the subsets, etc.) gives an improvement of accuracy. (4) When the quantity being evaluated through functional integration is the partition function, the device introduced by Feynman and Hibbs, of interchanging the functional integration with the integration of the Green's function over the equated initial and final configuration-space points, may be combined with the above techniques. This eliminates one integration in the terms of the expansion and seems to improve accuracy at the same time. The general series obtained is correlated with more conventional operator techniques of quantum-mechanical perturbation theory, in order to answer the perennial question, does the path-integral method bring with it anything that could not be derived by other methods? It is in some sense a Feynman-Dyson expansion of the Green's function, but one that is further modified mathematically in a way characteristic only of the path-integral point of view, and which, moreover, improves its accuracy. It thus appears unlikely that the result is merely one of standard type disguised as a functional integral result. Sample numerical calculations are given to assay the accuracy of the methods, which are shown to compare very favorably with the traditional approximation of finite subdivision of the time interval.
We establish a rather deep relation between statistical mechanics and quantum mechanics at an imaginary time. For the quantum field theory, this implies a relation between finite temperature quantum field theory in Minkowski space and quantum field theory in Euclidean space with one compact Euclidean time direction. The quantum mechanical language being a proper formulation of statistical physics as a molecular theory is basically the study of quantum equilibrium as well as non-equilibrium systems. A quantum system found in a thermodynamic equilibrium state with an external classical system (thermostat or thermal reservoir) may be characterized by fixed values of thermodynamic parameters of the system, say, for example, temperature T , volume V and number of particles N . In this way the total energy of the thermostat is assumed to be much greater than the energy of the system. The behavior of the quantum system may be assumed to be independent of the nature of the thermostat. The most convenient object of the quantum theory to describe such non-isolated systems is the density matrix since the partition function does not determine any local thermodynamic quantities. It is for this reason that we introduced quantum mechanical and quantum statistical density matrices which are necessary to do the investigation via path integration. This enables us to make a transition to the quasi-classical approximation of quantum statistical mechanics, which, in turn, enables us to evaluate the first quantum correction to the classical Boltzmann distribution function.
We have formulated the statistical mechanics in terms of the SSS matrix, which describes the scattering processes taking place in the thermodynamical system of interest. Such a formulation is necessary for studying the systems whose microscopic constituents behave according to the laws of relativistic quantum mechanics. Our result is a simple prescription for calculating the grand canonical potential of any gaseous system given the free-particle energies and SSS-matrix elements. When applied to a nonrelativistic gas, it gives a simple prescription for calculating all virial coefficients. Simplified relativistic gas models are considered as examples of application. A general form of the Levinson's Theorem for any number of particles follows immediately from our formalism. Its applications in statistical mechanics are briefly discussed.
We have formulated the statistical mechanics in terms of the SSS matrix, which describes the scattering processes taking place in the thermodynamical system of interest. Such a formulation is necessary for studying the systems whose microscopic constituents behave according to the laws of relativistic quantum mechanics. Our result is a simple prescription for calculating the grand canonical potential of any gaseous system given the free-particle energies and SSS-matrix elements. When applied to a nonrelativistic gas, it gives a simple prescription for calculating all virial coefficients. Simplified relativistic gas models are considered as examples of application. A general form of the Levinson's Theorem for any number of particles follows immediately from our formalism. Its applications in statistical mechanics are briefly discussed.