IRREVERSIBILITY AND GENERALIZED ENTROPIES

The time evolution of the Tsallis (T) and Renyi (R) entropies for a di screte state space was recently analyzed by Mariz [Phys. Lett. A 165 ( 1992) 409] and Ramshaw [Phys. Lett. A 175 (1993) 169] based on the mas ter equation. Here we perform a corresponding analysis for a continuou s state space x in which the probability distribution rho(x, t) obeys the generalized Liouville equation. For this purpose it is necessary t o formulate properly covariant generalizations of the T and R entropie s in terms of rho(x, t). We show that if the microscopic dynamics is r eversible in the Poincare-Lyapunov sense (i.e., D(x) = 0, where D(x) i s the covariant divergence of the flow velocity in state space) then b oth the T and R entropies are constant in time, just like the conventi onal entropy. The T and R entropies are therefore not intrinsically ir reversible. These results are obtained as special cases of a more gene ral result: if D(x) = 0 then dS/dt = 0 for any entropy functional S[rh o(x)] for which deltaS/deltarho(x) = f(rho(x)/gamma(x)), where gamma(x ) is the determinant of the metric tensor in state space and the funct ion f is arbitrary.