Canonical partition functions: ideal quantum gases, interacting classical gases, and interacting quantum gases

Abstract

In statistical mechanics, for a system with fixed number of particles, e.g., a finite-size system, strictly speaking, the thermodynamic quantity needs to be calculated in the canonical ensemble. Nevertheless, the calculation of the canonical partition function is difficult. In this paper, based on the mathematical theory of the symmetric function, we suggest a method for the calculation of the canonical partition function of ideal quantum gases, including ideal Bose, Fermi, and Gentile gases. Moreover, we express the canonical partition functions of interacting classical and quantum gases given by the classical and quantum cluster expansion methods in terms of the Bell polynomial in mathematics. The virial coefficients of ideal Bose, Fermi, and Gentile gases is calculated from the exact canonical partition function. The virial coefficients of interacting classical and quantum gases is calculated from the canonical partition function by using the expansion of the Bell polynomial, rather than calculated from the grand canonical potential.

Calculating eigenvalues of many-body systems from partition functions

“…In order to illustrate the influence of the quantum exchange interaction, we consider a D-dimensional system consisting of two bosons or two fermions. For the two-particle case, the exact canonical partition function is [31]…”

“…In statistical mechanics, many methods are developed for the calculation of partition functions. For example, the canonical partition function for quon statistics [24], general formulas for the canonical partition function of parastatistical systems [25], the canonical partition function of the freely jointed chain model [26], the partition function of the interacting calorons ensemble [27], the algorithm for computing the exact partition function of lattice polymer models [28], the exact partition function for the q-state Potts Model [29], the partition function for the antiferromagnetic Ising model and the hard-core models [30], and the canonical partition functions for different gaseous systems [31] are investigated.…”

A statistical mechanical approach to restricted integer partition functions

“…In this paper, first, by resorting to the canonical partition function of quantum ideal gases in statistical mechanics [14], we construct the generating function of restricted integer partition functions corresponding to ideal Bose, Fermi, and Gentile gases, respectively. The result shows that the generating functions for these restricted integer partition functions are symmetric functions which are invariant under the action of the permutation group and can be represented as linear combinations of the S-function which is an important class of symmetric functions [15,16].…”

“…The generating function is an effective way to calculate the partition function, especially in some cases that the generating function can be obtained relatively easy. Our starting point is the canonical partition function of these three quantum gases in statistical mechanics [14]. For a set {ε} = {ε 1 , ε 2 , .…”

“…Thus the number of microstates Ω q (E, N ) in the macrostate (N, E) is the number of representations of E in terms of N single-particle energies ε i which sum up to E and each ε i repeats no more than q times. By the definition of P The canonical partition function of an ideal Gentile gas is [14] Z…”