Kinetic Approach to Fractional Exclusion Statistics

Abstract: We show that the kinetic approach to statistical mechanics permits an elegant and efficient treatment of fractional exclusion statistics. By using the exclusion-inclusion principle recently proposed [Phys. Rev. E 49, 5103 (1994)] as a generalization of the Pauli exclusion principle, which is based on a proper definition of the transition probability between two states, we derive a variety of different statistical distributions interpolating between bosons and fermions. The Haldane exclusion principle and the Haldane-Wu fractional exclusion statistics are obtained in a natural way as particular cases. The thermodynamic properties of the statistical systems obeying the generalized exclusion-inclusion principle are discussed. PACS number(s): 05.20.-y, 05.30.-d, 73.40.Hm, 71.30.+h We show that the kinetic approach to statistical mechanics permits an elegant and efficient treatment of fractional exclusion statistics. By using the exclusion-inclusion principle recently proposed [Phys. Rev. E 49, 5103 (1994)] as a generalization of the Pauli exclusion principle, which is based on a proper definition of the transition probability between two states, we derive a variety of different statistical distributions interpolating between bosons and fermions. The Haldane exclusion principle and the Haldane-Wu fractional exclusion statistics are obtained in a natural way as particular cases. The thermodynamic properties of the statistical systems obeying the generalized exclusion-inclusion principle are discussed. PACS number(s): 05.20.-y, 05.30.-d, 73.40.Hm, 71.30.+h Typeset using REVTEX ∗e-mail: [email protected] 1 Following Haldane’s formulation [1] of a generalized Pauli exclusion principle, many papers have been recently devoted to the study of fractional exclusion statistics by interpolation of bosonic and fermionic distributions [2]. There is an intrinsic connection between these fractional statistics and the interpretation of the fractional quantum Halleffect [3] and anyonic physics [4,5]. Murthy and Shankar [6] generalized the Haldane statistics to infinite dimensions and showed [7] that the one dimensional bosons interacting through the twobody inverse square potential Vij = g(g − 1)(xi − xj) −2 of the Calogero-Sutherland model obeys fractional exclusion statistics in the sense of the Haldane interpretation. Using the Thomas-Fermi method, Sen and Bhaduri [8] considered the particle exclusion statistics for the Calogero-Sutherland model in the presence of an external, arbitrary one-body potential. Isakov [9] extended the Calogero-Sutherland model in the case of particles interacting through a generic two-body potential Vij and derived the related exclusion statistics. Rajagopal [10] obtained the von Neumann entropy per state of the Haldane exclusion statistics. Nayak and Wilczek [11] studied the low-temperature properties, the fluctuations and the duality of exclusions statistics. Murthy and Shankar [6] also computed, for quasi-particles, in the Luttinger model, the exclusion statistics parameter g related to the exchange statistical parameter α of the quantum phase e, and showed [7] that the parameter g can be completely determined by the second virial coefficient in the high temperature limit. Haldane defined the statistics g of a particle by g = − d N+∆N − d N ∆N , (1) where d N is the single-particle Hilbert-space dimension, when the coordinates of N − 1 particles are kept fixed. Thus the dimension of the Hilbert space for the single particle states is a finite and extensive quantity that depends on the number of particles in the system. One can choose this dimension as d N = K − g(N − 1) , (2) and the statistical weight or degeneracy factor is 2 W = (d N +N − 1)! N ! (d N − 1)! . (3) This equation is a simple interpolation between the number of ways of placing N identical bosons or fermions in K single-particle independent states confined to a finite region of matter. The expression of the mean occupation number n = N/K has been obtained in an implicit form by Wu [2]. n = 1 w(E) + g , (4) with w satisfying the relation w(E) [1 + w(E)] = e . (5) In the special cases g = 0 and g = 1, Eq.(4) yields, respectively, the Bose-Einstein (BE) and Fermi-Dirac (FD) distributions. Ilinski and Gunn [12] criticize the identity of relations (1) and (3) of Haldane and, using (2), derive a different W constructing a statistical mechanics which is not in complete agreement with Wu statistical mechanics [2]. Acharya and Swamy [13] and Polychronakos [14] have studied, in addition to the Haldane exclusion statistics, new fractional statistics with appealing features as positive probabilities and analytical expressions for all termodynamic quantities. We wish to recall the first work on intermediate statistics published by Gentile at the beginning of 1940 [15]. In a previous work [16] we have considered the kinetics of particles in a phase space of arbitrary dimensions D, obeying an exclusion-inclusion principle. We obtained the statistical distributions of the particle system as the stationary state of a non-linear kinetic equation. A crucial point of this formalism is the definition of the transition probability which can be written in various forms, containing the effects of the inclusion-exclusion principle through the distribution function n = n(t,v), which is the mean occupation number of the state v. Setting π(t,v → u) the transition probability from the state v to the state u, the evolution equation of the distribution function n(t,v) can be written as