Article
Irreversibility and Generalized Entropies
Physics Letters A
(1993)
Abstract
The time evolution of the Tsallis (T) and Renyi (R) entropies for a discrete state space was recently analyzed by Mariz [Phys. Lett. A 165 (1992) 409] and Ramshaw [Phys. Lett. A 175 (1993) 169] based on the master equation. Here we perform a corresponding analysis for a continuous state space χ in which the probability distribution ϱ(χ, t) obeys the generalized Liouville equation. For this purpose it is necessary to formulate properly covariant generalizations of the T and R entropies in terms of ϱ(χ, t). We show that if the microscopic dynamics is reversible in the Poincaré-Lyapunov sense (i.e., D(χ)=0, where D(χ) is the covariant divergence of the flow velocity in state space) then both the T and R entropies are constant in time, just like the conventional entropy. The T and R entropies are therefore not intrinsically irreversible. These results are obtained as special cases of a more general result: if D(χ)=0 then for any entropy functional S[ϱ(χ)] for which , where γ(χ) is the determinant of the metric tensor in state space and the function f is arbitrary.
Disciplines
Publication Date
April, 1993
DOI
10.1016/0375-9601(93)90821-G
Publisher Statement
At the time of publication John Ramshaw was affiliated with the Idaho National Engineering Laboratory.
Citation Information
J. D. Ramshaw, "Irreversibility and Generalized Entropies," Phys. Lett. A 175, 171 (1993).