Boltzmann Configurational Entropy Revisited in the Framework of Generalized Statistical Mechanics

As known, a method to introduce non-conventional statistics may be realized by modifying the number of possible combinations to put particles in a collection of single-particle states. In this paper, we assume that the weight factor of the possible configurations of a system of interacting particles can be obtained by generalizing opportunely the combinatorics, according to a certain analytical function $f_{\left\{\pi \right\}}\left(n\right)$ of the actual number of particles present in every energy level. Following this approach, the configurational Boltzmann entropy is revisited in a very general manner starting from a continuous deformation of the multinomial coefficients depending on a set of deformation parameters $\left\{\pi \right\}$. It is shown that, when $f_{\left\{\pi \right\}}\left(n\right)$ is related to the solutions of a simple linear difference–differential equation, the emerging entropy is a scaled version, in the occupational number representation, of the entropy of degree $(\kappa ,r)$ known, in the framework of the information theory, as Sharma–Taneja–Mittal entropic form.