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arXiv:cond-mat/9405017

Exclusion Statistics: Low-Temperature Properties, Fluctuations, Duality, Applications

Frank A. Wilczek, Chetan Nayak

Published May 6, 1994• 2 citations
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Abstract

We derive some physical properties of ideal assemblies of identical particles obeying generalized exclusion statistics. We discuss fluctuations, and in this connection point out a fundamental contrast to conventional quantum statistics. We demonstrate a duality relating the distribution of particles at statistics $g$ to the distribution of holes at statistics $1/g$. We suggest applications to Mott insulators.

DOI
10.48550/arxiv.cond-mat/9405017

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We derive some physical properties of ideal assemblies of identical particles obeying generalized exclusion statistics. We discuss fluctuations, and in this connection point out a fundamental contrast to conventional quantum statistics. We demonstrate a duality relating the distribution of particles at statistics ggg to the distribution of holes at statistics 1/g1/g1/g. We suggest applications to Mott insulators.
We derive some physical properties of ideal assemblies of identical particles obeying generalized exclusion statistics. We discuss fluctuations and in this connection point out a fundamental contrast to conventional quantum statistics. We demonstrate a duality relating the distribution of particles at statistics ggg to the distribution of holes at statistics 1g\frac{1}{g}g1​. We suggest an application to Mott insulators.
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.
After a very brief introduction to the concept of fractional exclusion statistics (FES), introduced by Haldane in Phys. Rev. Lett. 67, 937 (1991), I shall present some recent results regarding the basic properties of this statistics. These properties have been overlooked for almost two decades and because of that, the application of FES to concrete physical systems was, to a certain degree, arbitrary. Essentially, the FES is a generalization of the Pauli Exclusion Principle which allows one to describe systems of interacting particles (bosons or fermions) as systems of ideal particles, but which obey a more peculiar type of statistics. In the second part of the paper I shall present some application of FES by analyzing concrete systems of interacting particles.
I discuss ideal and interacting quantum gases obeying general fractional exclusion statistics. For systems with constant density of single-particle states, described in the mean field approximation, the entropy depends neither on the microscopic exclusion statistics, nor on the interaction. Such systems are called thermodynamically equivalent and I show that the microscopic reason for this equivalence is a one-to-one correspondence between the excited states of these systems. This provides a method, different from the bosonization technique, of transforming between systems of different exclusion statistics. In the last section the macroscopic aspects of this method are discussed. In appendix A, I calculate the fluctuation of the ground-state population of a condensed Bose gas in a grand-canonical ensemble and mean field approximation, while in appendix B I show a situation where although the system exhibits fractional exclusion properties on microscopic energy intervals, a rigorous calculation of the population of single-particle states reveals a condensation phenomenon. This also implies a malfunction of the usual and simplified calculation technique of the most probable statistical distributions.