On the Thermodynamics of the q-Particles

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).

Spectral actions for q-particles and their asymptotics

“…It is customary (e.g. [28], and further [24,10]) to perform the passage to the continuum as follows: for a gas with N particles, we simply make the replacement ε∈σ(H)…”

“…Since the properties of such free gases are now available for the general cases q ∈ [−1, 1], relative to q-particles or quons, after computing their grand partition function (cf. [10]), we carry out such an investigation for all cases, q = ±1 and q = 0 being the Bose/Fermi and Boltzmann cases, respectively. Indeed, for spectral actions arising from open systems made of such general free q-particles in thermal equilibrium, we compute the asymptotic expansion with respect to the natural cut-off associated to 1/β = k B T , T being the absolute temperature and k B ≈ 1.3806488 × 10 −23 JK −1 the Boltzmann constant.…”

“…Unfortunately, it is first seen in [32] and then in [10], that such a method to compute the grand partition function works only for the Bose/Fermi cases corresponding to q = ±1, see e.g. [6].…”

“…Concerning the remaining cases q ∈ (−1, 0) (0, 1), the partition function should be computed by a different method because it is completely unknown what should be the statistics enjoyed by such exotic particles. However, in [10] it was computed the reasonable partition function for all q ∈ [−1, 1], uniquely determined up to a multiplicative constant. Indeed, by using only the commutation relations (2.1) and imposing the Kubo-Martin-Schwinger condition in the grand canonical ensemble scheme, in [1] it was proven for such a free gas of quons, that the average population of particles for a fixed energy-level ε in thermodynamic equilibrium satisfies the natural extension…”

“…Therefore, the condensation takes place if and only if    d ≥ 2 for massless case , d ≥ 3 for massive case .It is then customary to separate the contribution corresponding to ground level ε = 0 from the remaining ones, see e.g [28],. and[10], Section 5. Therefore, we set a massless case when d = 1 and massive case when d = 1, 2 , ≥ 0 for massless case when d ≥ 2 and massive case when d ≥ 3…”