Tsallis Ensemble as an Exact Orthode

Abstract

We show that Tsallis ensemble of power-law distributions provides a mechanical model of nonextensive equilibrium thermodynamics for small interacting Hamiltonian systems, i.e., using Boltzmann's original nomenclature, we prove that it is an exact orthode. This means that the heat differential admits the inverse average kinetic energy as an integrating factor. One immediate consequence is that the logarithm of the normalization function can be identified with the entropy, instead of the q-deformed logarithm. It has been noted that such entropy coincides with Renyi entropy rather than Tsallis entropy, it is non-additive, tends to the standard canonical entropy as the power index tends to infinity and is consistent with the free energy formula proposed in [S. Abe et. al. Phys. Lett. A 281, 126 (2001)]. It is also shown that the heat differential admits the Lagrange multiplier used in non-extensive thermodynamics as an integrating factor too, and that the associated entropy is given by ordinary nonextensive entropy. The mechanical approach proposed in this work is fully consistent with an information-theoretic approach based on the maximization of Renyi entropy.

5 pages. Added connection with Renyi entropy

References in corpus (5)

• Interpretation of the nonextensitivity parameter $q$ in some applications of Tsallis statistics and Lévy distributions
• Thermodynamic Description of the Relaxation of Two-Dimensional Euler Turbulence Using Tsallis Statistics
• Nonextensive thermodynamic relations
• Temperature of nonextensive system: Tsallis entropy as Clausius entropy
• Is Tsallis thermodynamics nonextensive?

Cited by in corpus (8)

• Finite Bath Fluctuation Theorem
• Thermodynamics with generalized ensembles: The class of dual orthodes
• On the Limiting Cases of Nonextensive Thermostatistics
• Route from discreteness to the continuum for the non-logarithmic $q$-entropy
• The Physical Basis of the Gibbs-von Neumann entropy
• The q-exponential family in statistical physics
• Stochastic thermodynamics and fluctuation theorems for non-linear systems
• Verification of finite bath fluctuation theorem for a non-ergodic system