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arXiv:cond-mat/0702075 · Physical Review Letters

Comment on “Quantum Statistical Mechanics of an Ideal Gas with Fractional Exclusion Statistics in Arbitrary Dimension”

Wung-Hong Huang

Published Sep 14, 1998• 12 citations
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Abstract

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.

DOI
10.1103/physrevlett.81.2392

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It is mentioned that anyon thermodynamic potential Q(α,N)Q(\alpha, N)Q(α,N) could not be factorized in terms characteristic of the ideal boson α=0\alpha =0α=0 and fermion α=1\alpha =1α=1 gases by the relation Q(α,N)=(1−α)Q(0,Nb)+αQ(1,Nf)Q(\alpha, N) = (1-\alpha) Q(0, N_b)+ \alpha Q(1, N_f)Q(α,N)=(1−α)Q(0,Nb​)+αQ(1,Nf​) in which N=Nf+NbN=N_f +N_bN=Nf​+Nb​, that claimed in Phys. Rev. Lett. 78, 3233 (1997). Our analyses indicate that the thermodynamic quantities of anyon gas may be factorized as Q(α)=αQ(1)+(1−α)Q(0)Q(\alpha) = \alpha Q(1) + (1-\alpha) Q(0)Q(α)=αQ(1)+(1−α)Q(0) only in the two-dimension system.
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.
The quantum statistical mechanics of an ideal gas with fractional exclusion (i.e., Haldane-Wu) statistics in arbitrary dimensions is discussed. The general formulation for pressure and density of the system is obtained in a closed form in terms of the DDD-dimensional momentum representation, which can be regarded as a natural generalization of the classic results for Fermi and Bose gases. Using this, it is shown that ideal gases with fractional exclusion statistics can be regarded as composites of fermions and bosons, and that no condensation occurs at low temperature except for the pure boson case.
In a recent letter -Phys. Rev. Lett. 73, 3331 (1995)-, the conclusion was reached that, in the one-dimensional Calogero model, only the second virial coefficient is affected by the statistical parameter ααα, where ααα is related to the coupling constant κ/xij2κ/ x_{ij}^2κ/xij2​ of the Calogero interaction by κ=α(α+1)κ=α(α+1)κ=α(α+1). We argue that it is not so, i.e. all virial coefficients are affected, if the thermodynamic limit is properly taken.
In Phys. Rev. Lett. 67, 937 (1991) [1], Haldane introduced the fruitful concept of fractional exclusion statistics (FES). One of the most influential papers in which the thermodynamics of FES systems was deduced is Y.-S. Wu, Phys. Rev. Lett. 73, 922 (1994). Unfortunately, some important, but eventually subtle, properties of the exclusion statistics parameters were overlooked in the original paper [1] and in all the papers after that, including [2]. This omission makes the thermodynamics of FES systems inconsistent when mutual exclusion statistics is manifesting in the system. By this Comment I want to point-out this error--an error that persisted for such a long time--and to give the correct statistical mechanics interpretation of FES.