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arXiv:1905.09277

Multiple Exclusion Statistics: the kkk-mers problem

Julian Jose Riccardo

Published May 23, 2019• 1 citation
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Abstract

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 $G(n)$ is introduced to account for the dependence of the statistical exclusion on the occupation-number $n$. Results of thermodynamics and state occupation are shown for ideal lattice gases of linear particles of size $k$ ($k$-mers) where multiple exclusion occurs. Remarkable agreement is found with simulations from $k=2$ to $10$ where multiple exclusion dominates as $k$ increases.

DOI
10.48550/arxiv.1905.09277

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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.
A new distribution for systems of particles in equilibrium obeying the exclusion of correlated states is presented following Haldane's state counting. It relies upon an ansatz to deal with the multiple exclusion that takes place when the states accessible to single particles are spatially correlated and it can be simultaneously excluded by more than one particle. Haldane's statistics and Wu's distribution are recovered in the limit of noncorrelated states of the multiple exclusion statistics. In addition, an exclusion spectrum function G(n) is introduced to account for the dependence of the state exclusion on the occupation number n. The results of thermodynamics and state occupation are shown for ideal lattice gases of linear particles of size k (k-mers) where the multiple exclusion occurs. Remarkable agreement is found with grand-canonical Monte Carlo simulations from k=2 to 10 where the multiple exclusion dominates as k increases.
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.
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.
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.