“Correct entropy”in the analysis of complex systems: what is the consequence of rejecting the postulate of equal a priori probabilities?

Abstract

Entropy can clearly reflect the probability of the macrostate of a system only in case of validity of the basic postulate of statistical mechanics—the postulate of equal a priori probabilities of microstates. We have proved that for most non-physical macrosystems it loses its power, and the role of entropy has to be performedby a more general property, which has been found out in this study and called entropic divergence. In accordance with the principle of continuity, it includes the Boltzmann entropy. The properties of entropic divergence were considered through proving a number of theorems. This property has generally been found to have a minimal effect on the equilibrium state of the system. The conditional minimum of entropic divergence was disclosed as an example of formalism through which we have derived exponential and marginal hyperbolic distributions that take into account unequal a priori probabilities. The multiplicative form of the combined distribution allows consideration of the process of interaction between two or more macrosystems as the realization of a complex macroexperience. We have disclosed the possibility of using the research findings for analysing adaptive statistical interaction of aggregate macrosystems. The termsused in the studyfacilitate the development of quantitative methods of such analysis. We have given an example of the possibility of using this approach to compute the exponent in the power-law (hyperbolic) distributions.