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Bose-Einstein condensation and condensation of $q$-particles in equilibrium and non equilibrium thermodynamics: a new approach

In the setting of the principle of local equilibrium which asserts that the temperature is a function of the energy levels of the system, we exhibit plenty of steady states describing the condensation of free Bosons which are not in thermal equilibrium. The surprising facts are that the condensation can occur both in dimension less than 3 in configuration space, and even in excited energy levels. The investigation relative to non equilibrium suggests a new approach to the condensation, which allows an unified analysis involving also the condensation of $q$-particles, $-1\leq q\leq 1$, where $q=\pm1$ corresponds to the Bose/Fermi alternative. For such $q$-particles, the condensation can occur only if $0<q\leq1$, the case 1 corresponding to the standard Bose-Einstein condensation. In this more general approach, completely new and unexpected states exhibiting condensation phenomena naturally occur also in the usual situation of equilibrium thermodynamics. The new approach proposed in the present paper for the situation of $2^\text{nd}$ quantisation of free particles, is naturally based on the theory of the Distributions, which might hopefully be extended to more general cases