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arXiv:0906.4836

Revision of the fractional exclusion statistics

Dragoş-Victor Anghel

Published Jun 26, 2009• 4 citations
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Abstract

I discuss the concept of fractional exclusion statistics (FES) and I show that in order to preserve the thermodynamic consistency of the formalism, the exclusion statistics parameters should change if the species of particles in the system are divided into subspecies. Using a simple and intuitive model I deduce the general equations that have to be obeyed by the exlcusion statistics parameters in any FES system.

DOI
10.48550/arxiv.0906.4836

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I discuss the concept of fractional exclusion statistics (FES) and I show that in order to preserve the thermodynamic consistency of the formalism, the exclusion statistics parameters should change if the species of particles in the system are divided into subspecies. Using a simple and intuitive model I deduce the general equations that have to be obeyed by the exlcusion statistics parameters in any FES system.
I amend the concept of fractional exclusion statistics (FES) by introducing new conditions that make the formalism logically —and therefore physically— consistent. These conditions are imposed on the exclusion statistics parameters, defining how the parameters change if the species of particles in the system are divided into sub-species. Using a simple and intuitive model I deduce the general equations for the change of the exclusion statistics parameters at the redefinition of the particle species in any FES system.
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.
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.
I introduce an ansatz for the exclusion statistics parameters of fractional-exclusion-statistics (FES) systems and I apply it to calculate the statistical distribution of particles from both, bosonic and fermionic perspectives. To check the applicability of the ansatz, I calculate the FES parameters in three well-known models: a Fermi-liquid–type system, a one-dimensional quantum system described in the thermodynamic Bethe ansatz and quasi-particle excitations in a fractional quantum Hall (FQH) system. The FES parameters of the first two models satisfy the ansatz, whereas those of the third model, although close to the form given by the ansatz, represent an exception. In this case I also show that the general properties of the FES parameters, deduced elsewhere ( EPL , 87 (2009) 60009), are satisfied also by the parameters of the FQH liquid.
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 present briefly the concept of fractional exclusion statistics (FES) and I analyze two models of interacting particle systems. After I show that the models are equivalent by transforming one into the other by redefining some terms in the Hamiltonians, I calculate the heat capacity of the system and compare it with the heat capacity of an ideal Bose or Fermi gas. I show that under certain conditions the heat capacities may not be equal. I show that the interacting particle system may always be described as an ideal gas obeying FES and I show the method to construct such a gas.