UNCONVENTIONAL STATISTICAL MECHANICS I: General Theory and Derivation of a Nonequilibrium Ensemble in Renyi’s Approach

Unconventional Statistical Mechanics is introduced as a practical succedaneous to the conventional well established Boltzmann-Gibbs statistical mechanics, when in the use of the latter the researcher is impaired in his/her capacity for satisfying the Criterion of Sufficiency in statistics [Fisher, 1922], that is , a failure in the characterization of the system related to some aspect relevant to the given physical situation. To patch this limitation on the part of the observer, in order to make calculations of average values of observables and to obtain response functions, are introduced unconventional approaches. We present a detailed description of their construction, for, in continuation , resorting to the use of the particular case of Renyi's un-conventional statistics to build a nonequilibrium ensemble formalism. The unconventional distribution functions of fermions and bosons are obtained, and in the follow-up article we describe applications to the study of experimental results in semiconductor physics and in electro-chemistry involving nanometric scales and fractal-like structures, and some additional theoretical analysis is added. Statistical Mechanics of many-body systems has a long and successful history. The introduction of the concept of probability in physics originated mainly from the fundamental essay of Laplace [1], who incorporated and extended some earlier seminal ideas (see for example [2]). As well known, Statistical Mechanics attained the status of a well established discipline at the hands of Maxwell, Boltzmann, Gibbs, and others, and went through some steps related to changes, not in its fundamental structure, but just on the substrate provided by microscopic mechanics. Beginning with classical dynamics, statistical mechanics incorporated – as they went appearing in the realm of Physics – relativistic dynamics and quantum dynamics. Its application to the case of systems in equilibrium proceeded rapidly and with exceptional success: equilibrium statistical mechanics gave – starting from the microscopic level – foundations to Thermostatics, and the possibility to build a Response Function Theory. Applications to nonequilibrium systems began , mainly, with the case of local equilibrium in the linear regime following the pioneering work of Lars Onsager [3] (see also [4]). For systems arbitrarily deviated from equilibrium and governed by nonlinear kinetic laws, the derivation of an ensemble-like formalism proceeded at a slower pace than in the case of equilibrium, and somewhat cautiously, with a long list of distinguished scientists contributing to such development. It can be noticed that Statistical Mechanics gained in the fifties an alternative approach sustained on the basis of …