Bose—Einstein Condensation and Condensation of q-Particles in Equilibrium and Nonequilibrium Thermodynamics

In the setting of the principle of local equilibrium which asserts that the temperature is a function of energy levels of the system, we exhibit plenty of steady states describing 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 nonequilibrium suggests a new approach to the condensation which allows a unified analysis involving also condensation of q-particles, −1 ≤ q ≤ 1, where q = ±1 corresponds to the Bose/Fermi alternative. For such q-particles, it is shown that condensation can occur only if 0 < q ≤ 1, 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 even in the usual situation of equilibrium thermodynamics involving bosons. The new approach proposed in the present paper for the situation of second quantisation of free particles, is based on the theory of distributions, which might hopefully be extended to more general cases.