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

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 $S^{(q)}={\sum }_{i=1}^N(p_i-p_i^q)\mathrm{/}[q(q-1)]$ and Shiino’s generalized free energy $F^{(q)}=[{\sum }_{i=1}^N{p}_i(p_i\mathrm{/}{\pi }_i{)}^{q-1}-1]\mathrm{/}[q(q-1)]$, where ${\pi }_i$ is the stationary distribution. $d{F}^{(q)}\mathrm{/}dt=-f_d^{(q)}\le 0$ 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.