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Thermodynamics of non-abelian exclusion statistics

Huang, W H

Published Jan 1, 2000• 1 citation
OpenAlex

Abstract

The thermodynamic potential of ideal gases described by the simplest non-abelian statistics is investigated. I show that the potential is the linear function of the element of the abelian-part statistics matrix. Thus, the factorizable property in the Haldane (abelian) fractional exclusion shown by the author [W. H. Huang, Phys. Rev. Lett. 81, 2392 (1998)] is now extended to the non-abelian case. The complete expansion of the thermodynamic potential is also given.

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I discuss Haldane's concept of generalized exclusion statistics (1991 Phys. Rev. Lett. 67 937) and show that it leads to inconsistencies in the calculation of the particle distribution that maximizes the partition function. These inconsistencies appear when mutual exclusion statistics is manifested between different subspecies of particles in the system. In order to eliminate these inconsistencies, I introduce new mutual exclusion statistics parameters, which are proportional to the dimension of the Hilbert subspace on which they act. These new definitions lead to properly defined particle distributions and thermodynamic properties. In another paper ( Preprint 0710.0728), I show that the fractional exclusion statistics manifested in general systems with interaction have these, physically consistent, statistics parameters.

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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.

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Thermodynamic Properties of Generalized Exclusion Statistics

Hyun Seok Yang, Bum-Hoon Lee, Chanyong Park B

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.

arXiv:cond-mat/9606165•Jun 24, 1996•14 citations
The thermodynamic potential of ideal gases described by the simplest non-abelian statistics is investigated. I show that the potential is the linear function of the element of the abelian-part statistics matrix. Thus, the factorizable property in the Haldane (abelian) fractional exclusion shown by the author [W. H. Huang, Phys. Rev. Lett. 81, 2392 (1998)] is now extended to the non-abelian case. The complete expansion of the thermodynamic potential is also given.
The thermodynamics of two-species gases described by the Haldane exclusion statistics is investigated. I use the high-temperature perturbation to find the third Virial coefficient for the gas in an arbitrary dimension. The perturbative result indicates that the two-dimensional thermodynamic potential becomes the linear function of the element of the Haldane statistics matrix, if the matrix is symmetric and particle mass is species independent. I also present a nonperturbative method to prove this factorizable property and an exact expression of the thermodynamic potential is suggested.
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.
I discuss Haldane's concept of generalized exclusion statistics (1991 Phys. Rev. Lett. 67 937) and show that it leads to inconsistencies in the calculation of the particle distribution that maximizes the partition function. These inconsistencies appear when mutual exclusion statistics is manifested between different subspecies of particles in the system. In order to eliminate these inconsistencies, I introduce new mutual exclusion statistics parameters, which are proportional to the dimension of the Hilbert subspace on which they act. These new definitions lead to properly defined particle distributions and thermodynamic properties. In another paper ( Preprint 0710.0728), I show that the fractional exclusion statistics manifested in general systems with interaction have these, physically consistent, statistics parameters.
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.
We analytically calculate some thermodynamic quantities of an ideal ggg-on gas obeying generalized exclusion statistics. We show that the specific heat of a ggg-on gas (g≠0g \neq 0g=0) vanishes linearly in any dimension as T→0T \to 0T→0 when the particle number is conserved and exhibits an interesting dual symmetry that relates the particle-statistics at ggg to the hole-statistics at 1/g1/g1/g at low temperatures. We derive the complete solution for the cluster coefficients bl(g)b_l(g)bl​(g) as a function of Haldane's statistical interaction ggg in DDD dimensions. We also find that the cluster coefficients bl(g)b_l(g)bl​(g) and the virial coefficients al(g)a_l(g)al​(g) are exactly mirror symmetric (lll=odd) or antisymmetric (lll=even) about g=1/2g=1/2g=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 ggg-on gas with zero chemical potential shows thermodynamic properties similar to the photon statistics. We discuss some physical implications of our results.