H-theorem with generalized relative entropies and the Tsallis statistics

Generalized relative entropies with a real parameter q are proposed for Mar kovian stochastic processes, and the associated H-theorem is proved. The pr oposed entropies play the role of a Lyapunov function as the standard relat ive entropy known as the Kullback-Leibler divergence does. The H-theorem ca n 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 equi librium value.