Abstract
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 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 . It is shown that, when 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 known, in the framework of the information theory, as Sharma–Taneja–Mittal entropic form.
Highlights
Boltzmann entropy of a microcanonical ensemble is a measure of the number of possible microstates W[n] of a n-identical particles system, for a given macrostate characterized, in the simplest case, by the total particle number n and the total energy E
In the framework of the q-deformed statistical mechanics the only q-version of the probability entropy seems to capture all relevant statistical information contained in the q-version of the configurational entropy
With the aim of better understanding the emergence of generalized distributions often observed in natural and artificial complex systems, we have investigated a possible derivation of the configurational Boltzmann entropy by introducing a prescription for combinatorics dictated by an analytical function f{π}(x), dependent on a set of deformation parameters {π}, which takes into account possible statistical correlations among the monade of the system
Summary
Boltzmann entropy of a microcanonical ensemble is a measure of the number of possible microstates W[n] of a n-identical particles system, for a given macrostate characterized, in the simplest case, by the total particle number n and the total energy E. According to the maximal entropy principle, the most probable particle configuration is the one that maximizes the configurational entropy under the appropriate constraints given, in this case, by n and E As known, such statistics is characterized by an exponential tail that is typical of the Boltzmann–Gibbs distribution. Probability distribution functions observed in complex systems (like sociophysics, econophysics, biophysics and others) are plagued by the heavy tail that confers to the system an anomalous statistical behavior which differs significantly from that exponential characterizing the Boltzmann-Gibbs distribution Such anomalous statistical proprieties are embodied, in some sense, by the associated entropic form from which the (meta)-equilibrium distribution is derived according to the already cited maximal entropy principle.
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