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arXiv:cond-mat/0202274 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Combinatorial interpretation of Haldane-Wu fractional exclusion statistics

A. K. Aringazin, M. I. Mazhitov

Published Aug 26, 2002• 5 citations
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Abstract

Assuming that the maximal allowed number of identical particles in a state is an integer parameter, q, we derive the statistical weight and analyze the associated equation that defines the statistical distribution. The derived distribution covers Fermi-Dirac and Bose-Einstein ones in the particular cases q=1 and q--> infinity (n(i)/q-->1), respectively. We show that the derived statistical weight provides a natural combinatorial interpretation of Haldane-Wu fractional exclusion statistics, and present exact solutions of the distribution equation.

DOI
10.1103/physreve.66.026116

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Assuming that the maximal allowed number of identical particles in a state is an integer parameter, q, we derive the statistical weight and analyze the associated equation that defines the statistical distribution. The derived distribution covers Fermi-Dirac and Bose-Einstein ones in the particular cases q=1 and q--> infinity (n(i)/q-->1), respectively. We show that the derived statistical weight provides a natural combinatorial interpretation of Haldane-Wu fractional exclusion statistics, and present exact solutions of the distribution equation.
This paper is a revision of the combinatorics of fractional exclusion statistics (FES). More specifically, the following exact statement of the generalized Pauli principle is derived: for an NNN-particles system exhibiting FES of extended parameter g=q/rg=q/rg=q/r (qqq and rrr are co-prime integers such that 0<q≤r0 < q \leq r0<q≤r), we found that the allowed occupation number of a state is smaller than or equal to r−q+1r-q+1r−q+1 and \emph{not} to 1/g1/g1/g whenever q≠1q\neq 1q=1 and, moreover, the global occupancy shape (merely represented by a partition of NNN) is admissible if the number of states occupied by at least two particles is less than or equal to (N−1)/r(N-1)/r(N−1)/r (N≡1(modr)N \equiv 1 \pmod rN≡1(modr)). These counting rules allow distinguishing infinitely many families of FES systems depending on the parameter ggg and the size NNN. As an application of the main result, we study the probability distributions of occupancy configurations. For instance, the number of occupied states is found to be a hypergeometric random variable. Closed-form expressions for the expectation values and variances in the thermodynamic limit are presented. By way of comparison, we obtain parallel results regarding the Gentile intermediate statistics and demonstrate subtle similarities and contrasts with respect to FES.
We calculate the partition function of a gas of particles obeying Haldane exclusion statistics, using a definition of a Hilbert space having a `fractional dimension' and constructing appropriate coherent states. The fractional dimension is expressed though the form of the identity operator in the Hilbert space. We find that there many possible generalisations of the Pauli exclusion principle, with particular choices of the scalar product leading to consistency either with Haldane's original definition of the effective dimensionality of the Hilbert space or with the combinatorial procedure invoked by Haldane and Wu. We explicitly demonstrate that at low particle densities these definitions are equivalent.
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.