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arXiv:hep-th/9811166 · Modern Physics Letters A

GENERALIZED PARTITION FUNCTIONS, INTERPOLATING STATISTICS AND HIGHER VIRIAL COEFFICIENTS

P. F. BORGES, Henrique Boschi-Filho, C. FARINA

Published Jun 14, 1999• 5 citations
arXivHTMLPDF

Abstract

Starting from determinants at finite temperature obeying an intermediate boundary condition between the periodic (bosonic) and antiperiodic (fermionic) cases, we find results which can be mapped onto those obtained from anyons for the second virial coefficient. Using this approach, we calculate the corresponding higher virial coefficients and compare them with the results in the literature.

DOI
10.1142/s0217732399001310

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Starting from determinants at finite temperature obeying an intermediate boundary condition between the periodic (bosonic) and antiperiodic (fermionic) cases, we find results which can be mapped onto those obtained from anyons for the second virial coefficient. Using this approach, we calculate the corresponding higher virial coefficients and compare them with the results in the literature.
We have computed by a Monte Carlo method the fourth virial coefficient of free anyons, as a function of the statistics angle θ. It can be fitted by a four term Fourier series, in which two coefficients are fixed by the known perturbative results at the boson and fermion points. We compute partition functions by means of path integrals, which we represent diagramatically in such a way that the connected diagrams give the cluster coefficients. This provides a general proof that all cluster and virial coefficients are finite. We give explicit polynomial approximations for all path integral contributions to all cluster coefficients, implying that only the second virial coefficient is statistics dependent, as is the case for two-dimensional exclusion statistics. The assumption leading to these approximations is that the tree diagrams dominate and factorize.
Arovas and co-workers, in their well-known paper on the statistical mechanics of anyons [D. P. Arovas, R. Schrieffer, F. Wilczek, and A. Zee, Nucl. Phys. B251, 117 (1985)], found that the second virial coefficient of a two-dimensional gas of free anyons exhibits nonanalytic behavior as the statistical parameter is varied in the neighborhood of the Bose point. In this paper we consider the anyons to be interacting, via a two-body interaction U(\ensuremath{\Vert}r1{\mathbf{r}}_{1}r1​-r2{\mathbf{r}}_{2}r2​\ensuremath{\Vert}), where r2{\mathit{r}}^{2}r2U(r)\ensuremath{\rightarrow}0 as r\ensuremath{\rightarrow}0. Expanding the second virial coefficient in the vicinity of the Bose point, we show that the nonanalyticity is still present and independent of the particular form of U. Furthermore, we calculate explicitly, the second virial coefficient in the case of a short-range interaction.
The virial expansion method is applied within a harmonic approximation to an interacting N-body system of identical fermions. We compute the canonical partition functions for two and three particles to get the two lowest orders in the expansion. The energy spectrum is carefully interpolated to reproduce ground-state properties at low temperature and the noninteracting high-temperature limit of constant virial coefficients. This resembles the smearing of shell effects in finite systems with increasing temperature. Numerical results are discussed for the second and third virial coefficients as functions of dimension, temperature, interaction, and transition temperature between low- and high-energy limits.