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American Journal of Physics

A Note on the Partition Function for Systems of Independent Particles

Douglas Ike Ford

Published Feb 1, 1971• 15 citations
Publisher

Abstract

The N-particle Bose-Einstein and Fermi-Dirac canonical partition functions are expressed explicitly in terms of the single-particle partition function. The nature of the approximations made in using corrected Maxwell-Boltzmann statistics is thereby made more obvious. A recurrence relation is developed to facilitate calculations using the exact partition function.

DOI
10.1119/1.1986094

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The N-particle Bose-Einstein and Fermi-Dirac canonical partition functions are expressed explicitly in terms of the single-particle partition function. The nature of the approximations made in using corrected Maxwell-Boltzmann statistics is thereby made more obvious. A recurrence relation is developed to facilitate calculations using the exact partition function.
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We analyze the so-called classical limit of the quantum-mechanical canonical partition function. In order to do that, we define accurately the density matrix for symmetrized and antisymmetrized wave functions only (Bose-Einstein and Fermi-Dirac), and find an exact relation between them and the density matrix for non symmetrized functions (Maxwell-Boltzmann). Our results differ from the generally assumed in a numerical factor N!, for which we suggest a physical interpretation. We derive as well the reverse (and also exact) relation, to find the canonical partition function for non-symmetrized wave functions in terms of the corresponding function for fermions and bosons.
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.
A formulation is given whereby the grand partition function of a many-body system satisfying Bose-Einstein or Fermi-Dirac statistics is expressed in terms of certain UUU functions defined for the same system with Boltzmann statistics. It is then shown that these UUU functions can be evaluated in successive approximations in terms of a binary kernel BBB which can be computed from a solution of the two-body problem. The approach to the limit of infinite volume is studied. The example of a hard sphere interaction is discussed in some detail.
We calculate the canonical partition function Z_N for a system of N free particles obeying so-called `quon' statistics where q is real and satisfies |q|<1 by using simple counting arguments. We observe that this system is afflicted by the Gibbs paradox and that Z_N is independent of q. We demonstrate that such a system of particles obeys the ideal gas law and that the internal energy U ( and hence the specific heat capacity C_V ) is identical to that of a system of N free particles obeying Maxwell-Boltzmann statistics.
We calculate certain features of Bose–Einstein condensation in the ideal gas by using recurrence relations for the partition function. The grand canonical ensemble gives inaccurate results for certain properties of the condensate that are accurately provided by the canonical ensemble. Calculations in the latter can be made tractable for finite systems by means of the recurrence relations. The ideal one-dimensional harmonic Bose gas provides a particularly simple and pedagogically useful model for which detailed results are easily derived. An analysis of the Bose system via permutation cycles yields insight into the physical meaning of the recurrence relations.