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arXiv:2402.03429 · Physical review. E

Microscopic inclusion statistics in a discrete one-body spectrum

Stéphane Ouvry, Alexios P. Polychronakos

Published Jul 26, 2024• 1 citation
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Abstract

We present the microscopic formulation of inclusion statistics, a counterpoint to exclusion statistics in which particles tend to coalesce more than ordinary bosons. We derive the microscopic occupation multiplicities of one-body quantum states and show that they factorize into a product of clusters of neighboring occupied states with enhanced statistical weights. Applying this statistics to a one-dimensional gas of particles in a harmonic well leads to a Calogero-like n-body inclusion spectrum with interesting physical properties.

DOI
10.1103/physreve.110.014140

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We present the microscopic formulation of inclusion statistics, a counterpoint to exclusion statistics in which particles tend to coalesce more than ordinary bosons. We derive the microscopic occupation multiplicities of one-body quantum states and show that they factorize into a product of clusters of neighboring occupied states with enhanced statistical weights. Applying this statistics to a one-dimensional gas of particles in a harmonic well leads to a Calogero-like n-body inclusion spectrum with interesting physical properties.
We formulate and study the microscopic statistical mechanics of systems of particles with exclusion statistics in a discrete one-body spectrum. The statistical mechanics of these systems can be expressed in terms of effective single-level grand partition functions obeying a generalization of the standard thermodynamic exclusion statistics equation of state. We derive explicit expressions for the thermodynamic potential in terms of microscopic cluster coefficients and show that the mean occupation numbers of levels satisfy a nesting relation involving a number of adjacent levels determined by the exclusion parameter. We apply the formalism to the harmonic Calogero model and point out a relation with the Ramanujan continued fraction identity and appropriate generalizations.
We propose a type of quantum statistics which we call inclusion statistics, in which particles tend to coalesce more than ordinary bosons. Inclusion statistics is defined in analogy with exclusion statistics, in which statistical exclusion is stronger than in Fermi statistics, but now extrapolating beyond Bose statistics, resulting in statistical inclusion. A consequence of inclusion statistics is that the lowest space dimension in which particles can condense in the absence of potentials is d=2, unlike d=3 for the usual Bose-Einstein condensation. This reduction in the dimension happens for any inclusion stronger than bosons, and the critical temperature increases with stronger inclusion. Possible physical realizations of inclusion statistics involving attractive interactions between bosons may be experimentally achievable.
Following the definition of Haldane's exclusion statistics, we propose a new and simple representation of the spectrum and the state counting of the quantum Calogero model. We show that N non-interacting particles with statistical interactions confined in an external harmonic well give exactly the same energy spectrum and the same degeneracy as those of the N -particle quantum Calogero model. We also show that this picture exactly corresponds to the generalized ideal gas, which was introduced by Bernard and Wu.
We give two formulations of exclusion statistics (ES) using a variable number of bosonic or fermionic single-particle states which depends on the number of particles in the system. Associated bosonic and fermionic ES parameters are introduced and are discussed for FQHE quasiparticles, anyons in the lowest Landau level and for the Calogero-Sutherland model. In the latter case, only one family of solutions is emphasized to be sufficient to recover ES; appropriate families are specified for a number of formulations of the Calogero-Sutherland model. We extend the picture of a variable number of single-particle states to generalized ideal gases with statistical interaction between particles of different momenta. Integral equations are derived which determine the momentum distribution for single-particle states and distribution of particles over the single-particle states in the thermal equilibrium. \textcopyright{} 1996 the American Physical Society.
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.