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arXiv:cond-mat/0702070 · Physical review. B, Condensed matter

Statistics of anyon gas and the factorizable property of thermodynamic quantities

Wung-Hong Huang

Published Jun 15, 1996• 13 citations

Abstract

The statistical distribution function of anyon is used to find the eighth viral coefficient in the high-temperature limit and the equation of state in the low-temperature limit. The perturbative results indicate that the thermodynamic quantities, Q(\ensuremath{\alpha}), of the free anyon gas may be factorized in the terms characteristic of the ideal Bose (\ensuremath{\alpha}=0) and fermion (\ensuremath{\alpha}=1) gases, i.e., Q(\ensuremath{\alpha})=\ensuremath{\alpha}Q(1)+(1-\ensuremath{\alpha})Q(0). It is shown that the factorizable property of the thermodynamic quantities, to all orders, can be established from the property of the equivalence between the anyon statistics and statistics in a system with boson-fermion transmutation, which was found by us in a recent paper [W. H. Huang, Phys. Rev. E 51, 3729 (1995)]. \textcopyright{} 1996 The American Physical Society.

DOI: 10.1103/physrevb.53.15842

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The statistical distribution function of anyon is used to find the eighth viral coefficient in the high-temperature limit and the equation of state in the low-temperature limit. The perturbative results indicate that the thermodynamic quantities, Q(\ensuremath{\alpha}), of the free anyon gas may be factorized in the terms characteristic of the ideal Bose (\ensuremath{\alpha}=0) and fermion (\ensuremath{\alpha}=1) gases, i.e., Q(\ensuremath{\alpha})=\ensuremath{\alpha}Q(1)+(1-\ensuremath{\alpha})Q(0). It is shown that the factorizable property of the thermodynamic quantities, to all orders, can be established from the property of the equivalence between the anyon statistics and statistics in a system with boson-fermion transmutation, which was found by us in a recent paper [W. H. Huang, Phys. Rev. E 51, 3729 (1995)]. \textcopyright{} 1996 The American Physical Society.

It is mentioned that anyon thermodynamic potential

Q(\alpha, N)

could not be factorized in terms characteristic of the ideal boson

\alpha =0

and fermion

\alpha =1

gases by the relation

Q(\alpha, N) = (1-\alpha) Q(0, N_b)+ \alpha Q(1, N_f)

in which

N=N_f +N_b

, that claimed in Phys. Rev. Lett. 78, 3233 (1997). Our analyses indicate that the thermodynamic quantities of anyon gas may be factorized as

Q(\alpha) = \alpha Q(1) + (1-\alpha) Q(0)

only in the two-dimension system.

We propose a new generalized statistical mechanics for anyons based on a simple ansatz that has the correct familiar limiting cases. We derive the anyon equation of state from the grand partition function. We develop the virial expansion and obtain all the virial coefficients. Furthermore, we demonstrate how thermodynamic quantities such as the molar specific heat can readily be calculated. We discuss the properties of semions and also demonstrate that ideal non-relativistic anyons do not exhibit finite temperature condensation.

We analytically calculate some thermodynamic quantities of an ideal

g

-on gas obeying generalized exclusion statistics. We show that the specific heat of a

g

-on gas (

g \neq 0

) vanishes linearly in any dimension as

T \to 0

when the particle number is conserved and exhibits an interesting dual symmetry that relates the particle-statistics at

g

to the hole-statistics at

1/g

at low temperatures. We derive the complete solution for the cluster coefficients

b_l(g)

as a function of Haldane's statistical interaction

g

in

D

dimensions. We also find that the cluster coefficients

b_l(g)

and the virial coefficients

a_l(g)

are exactly mirror symmetric (

l

=odd) or antisymmetric (

l

=even) about

g=1/2

. In two dimensions, we completely determine the closed forms about the cluster and the virial coefficients of the generalized exclusion statistics, which exactly agree with the virial coefficients of an anyon gas of linear energies. We show that the

g

-on gas with zero chemical potential shows thermodynamic properties similar to the photon statistics. We discuss some physical implications of our results.

Aim Based on the generalized Bose-Einstein distribution function in nonextensive statistical mechanics,thermodynamical quantities of the two-dimensional Bose gas such as internal energy,entropy,free energy,heat capacity,etc.,are calculated. Methods The factorization approach is incorporated,and the ensuing statistics is derived. Results/Conclusion Comparing these results with those of the conventional Bose statistics,it is found that the thermodynamical quantities are of similar forms.In the limit q→1,the thermodynamical quantities of the generalized two-dimensional Bose gas will return to the results of the Bose statistics.