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arXiv:1808.00045

On the combinatorics of exclusion in Haldane fractional statistics

Nour-Eddine Fahssi

Published Jul 29, 2018• 0 citations
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Abstract

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 $N$-particles system exhibiting FES of extended parameter $g=q/r$ ($q$ and $r$ are co-prime integers such that $0 < q \leq r$), we found that the allowed occupation number of a state is smaller than or equal to $r-q+1$ and \emph{not} to $1/g$ whenever $q\neq 1$ and, moreover, the global occupancy shape (merely represented by a partition of $N$) is admissible if the number of states occupied by at least two particles is less than or equal to $(N-1)/r$ ($N \equiv 1 \pmod r$). These counting rules allow distinguishing infinitely many families of FES systems depending on the parameter $g$ and the size $N$. 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.

DOI
10.48550/arxiv.1808.00045

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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 show the possibility of describing fractional exclusion statistics (FES) as an occupancy process with global and local exclusion constraints. More specifically, using combinatorial identities, we show that FES can be viewed as “ball-in-box” models with appropriate weighting on the set of occupancy configurations (merely represented by a partition of the total number of particles). As a consequence, the following exact statement of the generalized Pauli principle is derived: for an [Formula: see text]-particles system exhibiting FES of extended parameter [Formula: see text] ([Formula: see text] and [Formula: see text] are co-prime integers such that [Formula: see text]), (1) the allowed occupation number of a state is less than or equal to [Formula: see text] and not to [Formula: see text] whenever [Formula: see text] and (2) the global occupancy shape is admissible if the number of states occupied by at least two particles is less than or equal to [Formula: see text] ([Formula: see text]). These counting rules allow distinguishing infinitely many families of FES systems depending on the parameter [Formula: see text] and the size [Formula: see text].
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.
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 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.
A new distribution for systems of particles obeying statistical exclusion of correlated states is presented following the Haldane's state counting. It relies upon a conjecture to deal with the multiple exclusion that takes place when the states available to single particles are spatially correlated and it can be simultaneously excluded by more than one particle. The Haldane's statistics [F. D. M. Haldane, Phys. Rev. Lett. 67, 937 (1991)] and Wu's distribution [Y.-S. Wu, Phys. Rev. Lett. 52, 2103 (1984)] are recovered in the limit of non-correlated states (constant statistical exclusion) of the multiple exclusion statistics. In addition, the exclusion spectrum function G(n)G(n)G(n) is introduced to account for the dependence of the statistical exclusion on the occupation-number nnn. Results of thermodynamics and state occupation are shown for ideal lattice gases of linear particles of size kkk (kkk-mers) where multiple exclusion occurs. Remarkable agreement is found with simulations from k=2k=2k=2 to 101010 where multiple exclusion dominates as kkk increases.