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Unified Framework for Generalized Statistics: Canonical Partition Function, Maximum Occupation Number, and Permutation Phase of Wave Function

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Abstract

Over the past decades, many kinds of generalized statistics are proposed through two approaches: (1) generalizing the permutation symmetry of the wave function and (2) generalizing the maximum occupation of the quantum state. Nevertheless, the connection between these two approaches is obscure. In this paper, we suggest a unified framework to describe various kinds of generalized statistics by using the representation theory of the permutation group and the unitary group. With this approach, we reveal the connection between the permutation phase and the maximum occupation number, through constructing a method to obtain the permutation phase and the maximum occupation number from the canonical partition function. We show that only bosonic and fermionic particles are completely indistinguishable under permutations. Particles obeying generalized statistics are not completely indistinguishable and thus are not quantum particles. Besides, we also give the following results: (1) providing a general formula of canonical partition functions of ideal N-particle gases who obey various kinds of generalized statistics, (2) revealing that the maximum occupation number is not sufficient to distinguish different kinds of generalized statistics, (3) specifying the permutation phases of wave functions for generalized statistics, and (4) proposing three new kinds of generalized statistics which seem to be the missing pieces in the puzzle.

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Acknowledgements

We are very indebted to Dr G. Zeitrauman for his encouragement. This work is supported in part by NSF of China under Grant No. 62106033, No. 11575125, and No. 11675119.

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Authors and Affiliations

  1. School of Engineering, Dali University, Dali, Yunnan, 671003, People’s Republic of China

    Chi-Chun Zhou

  2. Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin, 300071, People’s Republic of China

    Yu-Zhu Chen

  3. Department of Physics, Tianjin University, Tianjin, 300350, People’s Republic of China

    Chi-Chun Zhou, Yu-Zhu Chen & Wu-Sheng Dai

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  1. Chi-Chun Zhou
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  2. Yu-Zhu Chen
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  3. Wu-Sheng Dai
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Correspondence to Chi-Chun Zhou.

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Communicated by Hal Tasaki.

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Zhou, CC., Chen, YZ. & Dai, WS. Unified Framework for Generalized Statistics: Canonical Partition Function, Maximum Occupation Number, and Permutation Phase of Wave Function. J Stat Phys 186, 19 (2022). https://doi.org/10.1007/s10955-021-02865-4

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