Non-Euclidean-normed Statistical Mechanics

This analysis introduces a possible generalization of Statistical Mechanics within the framework of non-Euclidean metrics induced by the Lp norms. The internal energy is interpreted by the non-Euclidean Lp-normed expectation value of a given energy spectrum. The presented non-Euclidean adaptation of Statistical Mechanics involves finding the stationary probability distribution in the Canonical Ensemble by maximizing the Boltzmann–Gibbs and Tsallis entropy under the constraint of internal energy. The derived non-Euclidean Canonical probability distributions are respectively given by an exponential, and by a q-deformed exponential, of a power-law dependence on energy states. The case of the continuous energy spectrum is thoroughly examined. The Canonical probability distribution is analytically calculated for a power-law density of energy. The relevant non-Euclidean-normed kappa distribution is also derived. This analysis exposes the possible values of the q- or κ-indices, which are strictly limited to certain ranges, depending on the given Lp-norm. The equipartition of energy in each degree of freedom and the extensivity of the internal energy, are also shown. Surprisingly, the physical temperature coincides with the kinetically defined temperature, similar to the Euclidean case. Finally, the connection with thermodynamics arises through the well-known standard classical formalisms.