Temporal extensivity of Tsallis’ entropy and the bound on entropy production rate

Abstract

The Tsallis entropy, which is a generalization of the Boltzmann-Gibbs entropy, plays a central role in nonextensive statistical mechanics of complex systems. A lot of efforts have recently been made on establishing a dynamical foundation for the Tsallis entropy. They are primarily concerned with nonlinear dynamical systems at the edge of chaos. Here, it is shown by generalizing a formulation of thermostatistics based on time averages recently proposed by Carati [A. Carati, Physica A 348, 110 (2005)] that, whenever relevant, the Tsallis entropy indexed by q is temporally extensive: linear growth in time, i.e., finite entropy production rate. Then, the universal bound on the entropy production rate is shown to be 1/ absolute value (1-q). The property of the associated probabilistic process, i.e., the sojourn time distribution, determining randomness of motion in phase space is also analyzed.

Incidence of nonextensive thermodynamics in temporal scaling at Feigenbaum points

“…The second property relates to 'temporal extensivity' of entropy production [40] at critical attractors [2], [20], [22], [23]. That is, linear growth with time t of the entropy associated to an ensemble of trajectories.…”

“…where q is the entropic index, λ q (x in ) is the q-generalized Lyapunov coefficient, and exp q (x) ≡ [1 − (q − 1)x] −1/(q−1) is the q-exponential function. The second property relates to 'temporal extensivity' of entropy production [40] at critical attractors [2], [20], [22], [23]. That is, linear growth with time t of the entropy associated to an ensemble of trajectories.…”

On the fractal dimension of orbits compatible with Tsallis statistics

“…Obviously relation (13) shows that the mean number < m > of visited cells is equal to K if the argument of the exponentials is large, which typically occurs for large times (i.e. for large N ).…”

Compatibility between dynamics and Tsallis statistics

“…It turns out that F T s (n) doesn't have a closed expression in terms of known functions, except for some special cases, but expression for F T s (n) can however be given as a series expansion (details can be find in ref. [13], see also [14]).…”