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arXiv:2209.05884 · Entropy

On the Thermodynamics of the q-Particles

Fabio Ciolli, Francesco Fidaleo

Published Jan 20, 2022• 3 citations
arXivHTMLPDF

Abstract

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).

DOI
10.3390/e24020159

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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).
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.
Abstract. It is demonstrated that the statistical mechanical partition function can be used to construct various different forms of phase space distributions. This indicates that its structure is not restricted to the Gibbs–Boltzmann factor prescription which is based on counting statistics. With the widely used replacement of the Boltzmann factor by a generalised Lorentzian (also known as the q-deformed exponential function, where κ = 1∕|q − 1|, with κ, q ∈ R) both the kappa-Bose and kappa-Fermi partition functions are obtained in quite a straightforward way, from which the conventional Bose and Fermi distributions follow for κ → ∞. For κ ≠ ∞ these are subject to the restrictions that they can be used only at temperatures far from zero. They thus, as shown earlier, have little value for quantum physics. This is reasonable, because physical κ systems imply strong correlations which are absent at zero temperature where apart from stochastics all dynamical interactions are frozen. In the classical large temperature limit one obtains physically reasonable κ distributions which depend on energy respectively momentum as well as on chemical potential. Looking for other functional dependencies, we examine Bessel functions whether they can be used for obtaining valid distributions. Again and for the same reason, no Fermi and Bose distributions exist in the low temperature limit. However, a classical Bessel–Boltzmann distribution can be constructed which is a Bessel-modified Lorentzian distribution. Whether it makes any physical sense remains an open question. This is not investigated here. The choice of Bessel functions is motivated solely by their convergence properties and not by reference to any physical demands. This result suggests that the Gibbs–Boltzmann partition function is fundamental not only to Gibbs–Boltzmann but also to a large class of generalised Lorentzian distributions as well as to the corresponding nonextensive statistical mechanics.
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.
The statistics of q-oscillators, quons, and, to some extent, of anyons are studied and the basic differences among these objects are pointed out In particular, the statistical distributions for different bosonic and fermionic q-oscillators are found for their corresponding Fock space representations in the case when the Hamiltonian is identified with the number operator. In this case and for non-relativistic particles, the single-particle temperature Green function is defined with q-deformed periodicity conditions. The equations of state for nonrelativistic and ultrarelativistic bosonic q-gases in an arbitrary space dimension are found near Bose statistics, as well as that for an anyonic gas near Bose and Fermi statistics. The first corrections to the second virial coefficients are also evaluated.
By expanding the grand partition function, we obtain the expression for partition function of two dimensional free particles which is seen to be of the same form as that of one dimensional harmonic oscillators. For such systems it is shown that the difference in statistics does not appear in thermal behavior. When the energy levels of the particles are modified as ε n 1 , n 2 = Δ + b ( n 1 2 + n 2 2 ), Δ >0, the Bose condensation occurs and, if we set Δ =2 b , the condensation temperature qualitatively agrees with that of Osborne. The speciffic heat of free Bosons in a narrow box is calculated as the function of temperature, its width d being taken 7.1 Å and 14.2 Å. The results show that the maximum of the specific heat occurs at higher temperatures than that of the case d =∞, and its height becomes lower. The weak excitation of a Fermi system can be treated as excitons which obey Bose statistics. And we can show, by the method analogous to that of Ward and Wilks and Dingle, that the sound wave in a Fermi gas should be considered as the second sound, or the propagation of fluctuation in exciton distribution. Its propagation velocity is shown to become p0/3mp_{0}/\sqrt{3}mp0​/3​m, where p 0 is the momentum at the Fermi surface.
Thermodynamical properties of a model quon gas are reported. The quons considered are the objects obeying the deformed commutator aa†−qa†a=1 with −1⩽q⩽1. Although q=1 and q=−1 correspond respectively to boson and fermion statistics, it is not trivial to generate a quon gas which interpolates between the boson gas and the fermion gas without Pauli’s exclusion principle. The proposed model, interpolating between the two limits, can realize the Maxwell-Boltzmann limit at q=0 and simulate a free anyon gas. Under the restriction −1⩽q⩽1, the gas is deformed but bosonic in nature. It shows the Bose-Einstein condensation even in two dimensions.