We propose that the notion of Haldane's dimension and the corresponding statistics be treated in a probabilistic spirit. Motivated by the experience of dimensional regularization, the dimension of a space is defined as a trace of a diagonal ``unit operator'' where the diagonal matrix elements are not in unity in general but are probabilities to find a system in a given state. These probabilities are then uniquely defined by the rules of Haldane's statistics. The partition function for an ideal gas of the particles, a state-counting procedure, the entropy and a distribution function for this probabilistic definition are investigated and compared with previous works in the context of g particles. The corresponding creation and annihilation operators are introduced and Hamiltonians for interacting g particles are constructed. The stability of the thermodynamical properties of a gas of particles with exclusion statistics under the influence of hopping (or scattering) between states is investigated. \textcopyright{} 1996 The American Physical Society.
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.
We examine a new fractional statistics for arbitrary dimensions, based on an extension of Pauli’s exclusion principle, to allow for finite multi-occupancies of a single quantum state. By explicitly constructing the many-body Hilbert space, we obtain a new algebra of operators and a new thermodynamics. The new statistics is different from fractional exclusion statistics. For a single species of particles it is equivalent to parafermi statistics; but their algebras are very different for the case of multiple species.
The quantum statistical mechanics of an ideal gas with fractional exclusion (i.e., Haldane-Wu) statistics in arbitrary dimensions is discussed. The general formulation for pressure and density of the system is obtained in a closed form in terms of the
D-dimensional momentum representation, which can be regarded as a natural generalization of the classic results for Fermi and Bose gases. Using this, it is shown that ideal gases with fractional exclusion statistics can be regarded as composites of fermions and bosons, and that no condensation occurs at low temperature except for the pure boson case.
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.