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Journal of the Physical Society of Japan

Thermodynamics of the Exclusion Statistics Associated with the Quantum Calogero Model

Hideaki Ujino, Miki Wadati

Published May 15, 1996• 4 citations
Publisher

Abstract

The canonical ensemble is studied for the free quasiparticles that describe the thermodynamics of the quantum Calogero model with an arbitrary rational coupling constant. The free quasiparticle is identified with the generalized ideal gas with mutual statistical interactions. The energy distribution function is numerically computed. Properties of the thermal excitation are discussed.

DOI
10.1143/jpsj.65.1203

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The canonical ensemble is studied for the free quasiparticles that describe the thermodynamics of the quantum Calogero model with an arbitrary rational coupling constant. The free quasiparticle is identified with the generalized ideal gas with mutual statistical interactions. The energy distribution function is numerically computed. Properties of the thermal excitation are discussed.
Thermodynamics of the quantum Calogero model is studied by the microcanonical ensemble approach. We formulate a thermodynamic Bethe ansatz (TBA), whose reference state is the generalized ideal gas with fractional statistics (free a -on). Using the a -onic TBA, the free-quasiparticle picture for the Calogero model is identified as the free a -on. We calculate the thermodynamic potential and the distribution function for the Calogero model in the thermodynamic limit. The thermodynamic potential at thermal equilibrium can be understood by the Bernoulli equation. We also show that the thermodynamics of the Calogero model and the Sutherland model are the same except for the dispersion relations.
We discuss the relationship between the classical Lagrange theorem in mathematics and the quantum statistical mechanics and thermodynamics of an ideal gas of multispecies quasiparticles with mutual fractional exclusion statistics. First, we show that the thermodynamic potential and the density of the system are analytically expressed in terms of the language of generalized cluster expansions, where the cluster coefficients are determined from Wu's functional relations for describing the distribution functions of mutual fractional exclusion statistics. Second, we generalize the classical Lagrange theorem for inverting the one complex variable functions to that for the multicomplex variable functions. Third, we explicitly obtain all the exact cluster coefficients by applying the generalized Lagrange theorem.
In a recent letter -Phys. Rev. Lett. 73, 3331 (1995)-, the conclusion was reached that, in the one-dimensional Calogero model, only the second virial coefficient is affected by the statistical parameter ααα, where ααα is related to the coupling constant κ/xij2κ/ x_{ij}^2κ/xij2​ of the Calogero interaction by κ=α(α+1)κ=α(α+1)κ=α(α+1). We argue that it is not so, i.e. all virial coefficients are affected, if the thermodynamic limit is properly taken.
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.