Measure of the departure of the q-metastable stationary states from equilibrium

One great success of non-extensive statistical mechanics is that it offers a concrete statistical foundation for describing stationary states out of equilibrium characterized by kappa distributions, which are the analogues of Gaussian distributions in Boltzmann-Gibbs (BG) statistics. The Tsallis formalism offers a solid statistical foundation and provides a set of proven tools for understanding these distributions, including a consistent definition of temperature-the physical temperature, which characterizes the non-equilibrium stationary states. Here we develop a measure of the 'thermodynamic distance' of stationary states from equilibrium, called q-metastability, which fulfills specific conditions and is expressed in terms of the Tsallis entropic index, q, and the dimensionality of the system, f. We demonstrate its role in the spectrum-like arrangement of stationary states, so that all stationary states lie in a spectrum from q = 1 (equilibrium) to the maximum value of q, which specifies the furthest possible stationary state from equilibrium. We show that the constructed q-metastability measure characterizes the identity of each stationary state in a more consistent way than the q-index itself. Finally, as an application, we demonstrate the eventual transition of metastable states toward equilibrium.