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arXiv:math-ph/0604045 · Journal of Mathematical Physics

Random point fields for paraparticles of any order

Hiroshi Tamura, Keiichi R. Ito

Published Feb 1, 2007• 5 citations
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Abstract

Random point fields which describe gases consisting of paraparticles of any order p∊N are given by means of the canonical ensemble approach. The analysis for the cases of the parafermion gases are discussed in full detail and it is shown that the partition functions are pth power of that of the usual (i.e., p=1) fermion. The same is true for parabosons.

DOI
10.1063/1.2435987

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Random point fields which describe gases consisting of paraparticles of any order p∊N are given by means of the canonical ensemble approach. The analysis for the cases of the parafermion gases are discussed in full detail and it is shown that the partition functions are pth power of that of the usual (i.e., p=1) fermion. The same is true for parabosons.
Hartle and Taylor have shown that the cluster law imposes certain restrictions on the allowed symmetries of first-quantized systems of identical particles. We extend their results to show that all particles which are neither bosons nor fermions and which obey the cluster law can be divided into two classes, those of finite order, and those of infinite order. For every positive integer ppp there are two types of finite-order particle, which we call parabosons and parafermions of order ppp. Ordinary bosons and fermions can be fitted into this scheme as particles of order 1. We conjecture that the finite-order particles can be identified with the parafermions and parabosons of the second-quantized, parafield theory. Infinite-order particles would seem to have no analog in the second-quantized theory, as presently formulated.
This study aims to determine the thermodynamic properties of the parastatistics system of order two. The thermodynamic properties to be searched include the Grand Canonical Partition Function (GCPF) Z, and the average number of particles N. These parastatistics systems is in a more general form compared to quantum statistical distribution that has been known previously, i.e.: the Fermi-Dirac (FD) and Bose-Einstein (BE). Starting from the recursion relation of grand canonical partition function for parastatistics system of order two that has been known, recuresion linkages for some simple thermodynamic functions for parastatistics system of order two are derived. The recursion linkages are then used to calculate the thermodynamic functions of the model system of identical particles with limited energy levels which is similar to the harmonic oscillator. From these results we concluded that from the Grand Canonical Partition Function (GCPF), Z, the thermodynamics properties of parastatistics system of order two (paraboson and parafermion) can be derived and have similar shape with parastatistics system of order one (Boson and Fermion). The similarity of the graph shows similar thermodynamic properties. Keywords: parastatistics, thermodynamic properties
We show that for any potential with a discrete energy spectrum, the well-known interpolation between ideal boson and fermion partition functions at discrete values of ξ=−1/mξ=-1/mξ=−1/m yielded zero temperature ground state energies corresponding to mmm fermions occupying a single quantum state. The grand canonical partition function in this case can be a result from genuine parastatistics.