Thermodynamics of Markov Processes with Non-extensive Entropy and Free Energy

Hong Qian

Published 2010-05-07, updated 2011-02-02Version 2

Parallel to the recent presented complete thermodynamic formalism for master equation systems, we show that a "thermodynamic" theory can also be developed based on Tsallis' generalized entropy $S_q and Shiino's generalized free energy F_q which depends on \pi_i, the stationary distribution of the master equation. $dF_q/dt=-f_d\le 0$ and it is zero iff the system is in its stationary state. $dS_q/dt = f_d-Q_{ex}$ where $Q_{ex}$ characterizes the heat exchange. For systems approaching equilibrium with detailed balance, $f_d$ is the product of Onsager's thermodynamic flux and force. However, it is discovered that the Onsager's force is non-local. This is a consequence of the particular transformation invariance for zero energy of Tsallis' statistics.