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The canonical partition function for quons

Goodison, J W, David J. Toms

Published Oct 13, 1994• 6 citations
OpenAlex

Abstract

We calculate the canonical partition function Z_N for a system of N free particles obeying so-called `quon' statistics where q is real and satisfies |q|<1 by using simple counting arguments. We observe that this system is afflicted by the Gibbs paradox and that Z_N is independent of q. We demonstrate that such a system of particles obeys the ideal gas law and that the internal energy U ( and hence the specific heat capacity C_V ) is identical to that of a system of N free particles obeying Maxwell-Boltzmann statistics.

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We calculate the canonical partition function Z_N for a system of N free particles obeying so-called `quon' statistics where q is real and satisfies |q|<1 by using simple counting arguments. We observe that this system is afflicted by the Gibbs paradox and that Z_N is independent of q. We demonstrate that such a system of particles obeys the ideal gas law and that the internal energy U ( and hence the specific heat capacity C_V ) is identical to that of a system of N free particles obeying Maxwell-Boltzmann statistics.
We consider the statistical mechanics of systems of particles satisfying the qqq-commutation relations recently proposed by Greenberg and others. We show that although the commutation relations approach Bose (Fermi) relations for $q\ensuremath{\rightarrow}1$ ($q\ensuremath{\rightarrow}\ensuremath{-}1$), the partition functions of free gases are independent of qqq in the range $\ensuremath{-}1<q<1$. The partition functions exhibit Gibbs' paradox in the same way as a classical gas without a correction factor 1N!\frac{1}{N}!N1​! for the statistical weight of the NNN-particle phase space; i.e., the statistical mechanics does not describe a material for which entropy, free energy, and particle number are extensive thermodynamical quantities.
Since the grand partition function Zq for the so-called q-particles (i.e., quons), q∈(−1,1), cannot be computed by using the standard 2nd quantisation technique involving the full Fock space construction for q=0, and its q-deformations for the remaining cases, we determine such grand partition functions in order to obtain the natural generalisation of the Plank distribution to q∈[−1,1]. We also note the (non) surprising fact that the right grand partition function concerning the Boltzmann case (i.e., q=0) can be easily obtained by using the full Fock space 2nd quantisation, by considering the appropriate correction by the Gibbs factor 1/n! in the n term of the power series expansion with respect to the fugacity z. As an application, we briefly discuss the equations of the state for a gas of free quons or the condensation phenomenon into the ground state, also occurring for the Bose-like quons q∈(0,1).
I present briefly the concept of fractional exclusion statistics (FES) and I analyze two models of interacting particle systems. After I show that the models are equivalent by transforming one into the other by redefining some terms in the Hamiltonians, I calculate the heat capacity of the system and compare it with the heat capacity of an ideal Bose or Fermi gas. I show that under certain conditions the heat capacities may not be equal. I show that the interacting particle system may always be described as an ideal gas obeying FES and I show the method to construct such a gas.
Bose–Einstein and Fermi–Dirac statistics versus Boltzmann statistics It was remarked in Chapter 2, §2.1, that there are circumstances in which the “Boltzmann statistics” for an ideal gas – the approximation in Eq. (2.6), in which the total partition function is decomposed into a product of single-molecule partition functions, with division by N 1 ! N 2 ! … to correct for particle identity – is inaccurate even if the gas is still ideal (non-interacting particles). That happens when the number of single-particle states within the energy kT of the ground state is no longer very much greater than the number of particles. (Recall Fig. 2.1 and the accompanying discussion.) We noted that when the Boltzmann statistics is no longer an adequate approximation one must distinguish the particles as fermions, no two of which can be in the same single-particle state (counting states of different spin as different), or as bosons, for which there is no restriction on multiple occupancy of the singleparticle states. Fermions are particles of half-odd-integer spin quantum number; they could be composite particles consisting of an odd number of spin-½ particles. Protons, neutrons, and electrons, all of which are of spin ½, are fermions. Bosons are particles of integer spin; they could be spinless, or of spin 1, etc., or could be composites consisting of an even number of fermions. The nuclei of helium atoms of mass number 4 consist of two protons and two neutrons, and the neutral atom has two electrons outside the nucleus; 4 He atoms are bosons.