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Physical Review

Quantum Statistics of Nonideal Systems

Frank H. Stillinger, John G. Kirkwood

Published Apr 15, 1960• 2 citations
Publisher

Abstract

A new cluster development for the logarithm of the grand partition function of a system of interacting particles is derived. The leading term in this expansion is the pressure exerted by an ideal Bose or Fermi gas at the same temperature and absolute activity $Z$ as the actual system. Succeeding terms involve quantum cluster integrals which themselves depend upon $Z$, unlike their classical analogs. The definition of these cluster integrals follows in a natural fashion using techniques illustrated by construction (in closed form) of the successive $Z$ derivatives of the Bose and Fermi ideal gas grand partition functions. It is not possible (except in the classical limit) to eliminate $Z$ explicitly between the pressure and density series, so that the equation of state must remain in parametric form.

DOI
10.1103/physrev.118.361

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A new cluster development for the logarithm of the grand partition function of a system of interacting particles is derived. The leading term in this expansion is the pressure exerted by an ideal Bose or Fermi gas at the same temperature and absolute activity ZZZ as the actual system. Succeeding terms involve quantum cluster integrals which themselves depend upon ZZZ, unlike their classical analogs. The definition of these cluster integrals follows in a natural fashion using techniques illustrated by construction (in closed form) of the successive ZZZ derivatives of the Bose and Fermi ideal gas grand partition functions. It is not possible (except in the classical limit) to eliminate ZZZ explicitly between the pressure and density series, so that the equation of state must remain in parametric form.
A systematic generalization of the Mayer cluster integral theory has been developed to deal with the quantum statistics of interacting particles. The grand partition function appears in a natural way and the cluster integrals are integrals over propagators which are derived from the Green's function solution of the Bloch equation (which follows from the Schroedinger equation by replacing it/ℏ by β = 1/kT). Every cluster integral can be represented by a hybrid of a Mayer graph and a Feynman diagram in (β, r) space. The generalization of classical ring cluster integrals has been analyzed. It is shown that in the case of the electron gas the classical limit of the contribution of these integrals to the grand partition function yields the Debye-Huckel theory while the low temperature limit leads to the Gell-Mann—Brueckner equation for the correlation energy of the ground state. A prescription is given for the construction of the cluster integral associated with any given diagram.
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed
The quantum statistical mechanics of an ideal relativistic Bose gas of massive particles is discussed. The thermodynamic functions of the system are obtained from the exact expression for the logarithm of the grand partition function. The correct procedure for carrying out the non-relativistic and ultra-relativistic limits is presented. The limitations and errors of previous calculations which relied on the binomial expansion of the density of states are pointed out. It is shown that the behaviour of the system in the limit of zero temperature is dominated by the non-relativistic effects.
Recursion formulas are derived for Fermi-Dirac and Bose-Einstein statistical partition functions for systems where the energy can be written as a sum of one particle energies. Thus their exact partition functions can be calculated for any N. We consider a system of N particles for which the energy can be written as a sum of one-particle energies such as ideal gases or simple shell models. We aim at a recursion of Z(N)/ in terms of Z for subsystems. The idea may be seen as a vague analogy to cluster expansion methods.
It is well known that the Boltzmann partition function for noninteracting particles factors into a product of N single-particle partition functions. We show here that, in much the same way, the logarithm of the grand partition function of quantum statistical mechanics for ideal gases is represented by a contour integral containing the single-particle partition function. The unification achieved in this way is illustrated by applications to field-free nonrelativistic particles, collections of relativistic particles, systems in the presence of a uniform mangetic field, and inhomogeneous systems in equilibrium. Both Fermi and Bose statistics are encompassed in the treatment.
The quantum statistical mechanics of an ideal gas with fractional exclusion (i.e., Haldane-Wu) statistics in arbitrary dimensions is discussed. The general formulation for pressure and density of the system is obtained in a closed form in terms of the DDD-dimensional momentum representation, which can be regarded as a natural generalization of the classic results for Fermi and Bose gases. Using this, it is shown that ideal gases with fractional exclusion statistics can be regarded as composites of fermions and bosons, and that no condensation occurs at low temperature except for the pure boson case.