Thermodynamics and time-average

Abstract

For a dynamical system far from equilibrium, one has to deal with empirical probabilities defined through time-averages, and the main problem is then how to formulate an appropriate statistical thermodynamics. The common answer is that the standard functional expression of Boltzmann-Gibbs for the entropy should be used, the empirical probabilities being substituted for the Gibbs measure. Other functional expressions have been suggested, but apparently with no clear mechanical foundation. Here it is shown how a natural extension of the original procedure employed by Gibbs and Khinchin in defining entropy, with the only proviso of using the empirical probabilities, leads for the entropy to a functional expression which is in general different from that of Boltzmann--Gibbs. In particular, the Gibbs entropy is recovered for empirical probabilities of Poisson type, while the Tsallis entropies are recovered for a deformation of the Poisson distribution.

8 pages, LaTex source. Corrected some misprints

Cited by in corpus (6)

• Transition records of stationary Markov chains
• On the definition of temperature using time--averages
• Temporal extensivity of Tsallis' entropy and the bound on entropy production rate
• Emergence of $q$-statistical functions in a generalized binomial distribution with strong correlations
• Approach to equilibrium via Tsallis distributions in a realistic ionic--crystal model and in the FPU model
• A generalized thermodynamics for power-law statistics