H-Theorem with Generalized Relative Entropies and the Tsallis Statistics

Generalized relative entropies with a real parameter q are proposed for Markovian stochastic processes, and the associated H-theorem is proved. The proposed entropies play the role of a Lyapunov function as the standard relative entropy known as the Kullback-Leibler divergence does. The H-theorem can be applied to Tsallis statistics, and it is shown that the nonequilibrium free energy, consistently defined based on the generalized entropy in the framework of Tsallis statistics, monotonically decreases to attain its equilibrium value.