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≤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=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≡1(modr)). 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.
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.
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.
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.
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.