Calculating statistical distributions from operator relations: the statistical distributions of various intermediate statistics

Abstract

In this paper, we give a general discussion on the calculation of the statistical distribution from a given operator relation of creation, annihilation, and number operators. Our result shows that as long as the relation between the number operator and the creation and annihilation operators can be expressed as $a^{†}b=\Lambda \left(N\right)$ or $N={\Lambda }^{-1}\left(a^{†}b\right)$, where $N$, $a^{†}$, and $b$ denote the number, creation, and annihilation operators, i.e., $N$ is a function of quadratic product of the creation and annihilation operators, the corresponding statistical distribution is the Gentile distribution, a statistical distribution in which the maximum occupation number is an arbitrary integer. As examples, we discuss the statistical distributions corresponding to various operator relations. In particular, besides Bose-Einstein and Fermi-Dirac cases, we discuss the statistical distributions for various schemes of intermediate statistics, especially various $q$-deformation schemes. Our result shows that the statistical distributions corresponding to various $q$-deformation schemes are various Gentile distributions with different maximum occupation numbers which are determined by the deformation parameter $q$. This result shows that the results given in much literature on the $q$-deformation distribution are inaccurate or incomplete.

20 pages, no figure

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• A New Method for Derivation of Statistical Weight of the Gentile Statistics
• Unified framework for generalized quantum statistics: canonical partition function, maximum occupation number, and permutation phase of wave function
• Multiparticle distributions and intercepts of $r$-particle correlation functions in the symmetric Tamm-Dancoff type $q$-Bose gas model
• A group method solving many-body systems in intermediate statistical representation