Statistical mechanical foundations of power-law distributions

The foundations of the Boltzmann–Gibbs (BG) distributions broadly fall into (i) probabilistic approaches based on the principle of equal a priori probability, the central limit theorem, or the state density considerations and (ii) the Gibbs–Jaynes maximum entropy principle. A minimal set of requirements on each of these are the function space, the counting algorithm, and “additivity” property of the entropy. In the past few decades, a class of complex systems, which are not necessarily in thermodynamic equilibrium (e.g., glasses), have been found to display power-law distributions, which are not describable by the traditional methods. Here, parallels to all the inquiries underlying the BG theory are given for the power-law distributions. In particular, a different function space is employed and additivity of the entropy is discarded. The requirement of stability identifies the entropy proposed by Tsallis. From this, a generalized thermodynamic description of such systems in quasi-equilibrium states is developed.