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

Abstract

Statistical thermodynamics of small systems shows dramatic differences from normal systems. Parallel to the recently presented steady-state thermodynamic formalism for master equation and Fokker-Planck equation, we show that a ``thermodynamic'' theory can also be developed based on Tsallis' generalized entropy $S^{(q)}=\sum_{i=1}^N(p_i-p_i^q)/[q(q-1)]$ and Shiino's generalized free energy $F^{(q)}=[\sum_{i=1}^Np_i(p_i/\pi_i)^{q-1}-1]/[q(q-1)]$, where $\pi_i$ is the stationary distribution. $dF^{(q)}/dt=-f_d^{(q)}\le 0$ and it is zero iff the system is in its stationary state. $dS^{(q)}/dt-Q_{ex}^{(q)} = f_d^{(q)}$ where $Q_{ex}^{(q)}$ characterizes the heat exchange. For systems approaching equilibrium with detailed balance, $f_d^{(q)}$ 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.