In statistical mechanics, for a system with fixed number of particles, e.g.,a finite-size system, strictly speaking, the thermodynamic quantity needs to becalculated in the canonical ensemble. Nevertheless, the calculation of thecanonical partition function is difficult; even the canonical partitionfunction of ideal Bose and Fermi gases cannot be obtained exactly. In thispaper, based on the mathematical theory of the symmetric function and the Bellpolynomial, we suggest a method for the calculation of the canonical partitionfunction of ideal quantum gases, including ideal Bose, Fermi, and Gentilegases. Moreover, we also reveal that the canonical partition functions ofinteracting classical and quantum gases given by the classical and quantumcluster expansion methods are indeed the Bell polynomial in mathematics. Thevirial coefficients of ideal Bose, Fermi, and Gentile gases is calculated fromthe exact canonical partition function. The virial coefficients of interactingclassical and quantum gases is calculated from the canonical partition functionby using the expansion of the Bell polynomial, rather than calculated from thegrand canonical potential.
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