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AIP conference proceedings

Combinatorial aspects of exclusion and parastatistics

S. Chaturvedi

Published Jan 1, 2003• 0 citations
Publisher

Abstract

Combinatorial aspects of all statistics based on the permutation group are analyzed by imposing the requirements of indistinguishability in the permutation group sense on the Hilbert space describing N identical particles. Compact expressions for the grand canonical partition functions are given wherever possible. The theory of symmetric functions is found to play a significant role in this development. An analysis of the semion statistics of Haldane is also presented from this perspective together with some recent developments in the field of exclusion statistics.

DOI
10.1063/1.1639585

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Combinatorial aspects of all statistics based on the permutation group are analyzed by imposing the requirements of indistinguishability in the permutation group sense on the Hilbert space describing N identical particles. Compact expressions for the grand canonical partition functions are given wherever possible. The theory of symmetric functions is found to play a significant role in this development. An analysis of the semion statistics of Haldane is also presented from this perspective together with some recent developments in the field of exclusion statistics.
A general formula for the canonical partition function for a parastatistical system of any order is derived. The formula expresses the canonical partition functions for these in terms of sums of Schur functions.The only hitherto known result due to Suranyi [Phys. Rev. Lett. 65, 2329 (1990)] for parasystems of order two is obtained as a special case. Our results apply not only to parastatistics but to all statistics that can be defined on the basis of the permutation group, including those for which no simple definition in terms of the algebra of creation and annihilation operators is possible. \textcopyright{} 1996 The American Physical Society.
Abstract The empirical rule that systems of identical particles always obey either Bose or Fermi statistics is customarily imposed on the theory by adding it to the axioms of nonrelativistic quantum mechanics, with the result that other statistical behaviors are excluded a priori. A more general approach is to ask what other many-particle statistics are consistent with the indistinguishability of identical particles. This strategy offers a way to discuss possible violations of the Pauli Exclusion Principle, and it leads to some interesting issues related to preparation of states and a superselection rule arising from invariance under the permutation group.
The connection between the exclusion principle and the principle of indistinguishability of identical particles in quantum mechanics is discussed. It is shown with the aid of a density matrix defined for states with arbitrary permutation symmetry that the particles are indistinguishable only for one-dimensional representations of the permutation group (symmetrical and antisymmetrycal), and are distinguishable in all the states that are degenerate in the permutations. It is demonstrated by the same token that the exclusion principle follows from the principle of indistinguishability of identical particles and the Pauli theorem on the connection between spin and statistics.
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 chapter follows on from the previous chapter in quantum statistical mechanics but specialising on systems with identical particles. Using Gibbs prescritpion on generic states from Chapter 3, the occupation number representation is introduced. Constraints imposed on statistics by irreducible representations of the permutation group are discussed. These group-theoretic considerations are used to justify the use of Gentile’s parastatistics. Fermions and bosons are introduced as special cases of Gentile’s statistics, corresponding to the trivial representation of the permutation group for bosons, and the sign representation of the permutation group for fermions. Basic applications to fermions and bosons is given, including the Fermi–Dirac and Bose–Einstein statistics. A detailed expose of why photons are said to have zero chemical potential is also proposed.
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.