Abstract
Since the grand partition function for the so-called q-particles (i.e., quons), , cannot be computed by using the standard 2nd quantisation technique involving the full Fock space construction for , 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 . We also note the (non) surprising fact that the right grand partition function concerning the Boltzmann case (i.e., ) can be easily obtained by using the full Fock space 2nd quantisation, by considering the appropriate correction by the Gibbs factor 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 .
Highlights
We indicate such grand partition functions as Z±1, where ±1 correspond to the Bose/Fermi alternative
−1 < q < 0, and all such grand partition functions are well defined in their own domain involving the activity z, and we have the following estimate for the Zq
It is well known that the main ingredient to deal with the thermodynamics of macroscopic systems in terms of statistical mechanics is the grand partition function, which in the Fermi/Bose cases can be computed by the standard techniques of second quantisation
Summary
Concerning the grand partition function Z, it comes by considering open systems in thermodynamic equilibrium at inverse temperature β and chemical potential μ It is computed as standard for Fermi and Bose particles with the use of the symmetric and totally anti-symmetric (due to Pauli exclusion principle) Fock spaces F± (H), see [28], Section 5.2.1. It might be natural to use the the so-called full Fock space F(H) and the grand canonical Hamiltonian K := dΓ( H ) − μN, being dΓ( H ) the second quantized of the operator H and N the number operator, as for the computation of Z±1 in the Bose and Fermi cases, see e.g., [28], Section 5.2.1 This computation is precisely what was done in [17], Formula (11), obtaining.
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