Thermodynamics of Markov processes with nonextensive 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)}=∑_{i=1}^{N}(p_{i}-p_{i}^{q})/[q(q-1)] and Shiino's generalized free energy F^{(q)}=[∑_{i=1}^{N}p_{i}(p_{i}/π_{i})^{q-1}-1]/[q(q-1)], where π_{i} is the stationary distribution. dF^{(q)}/dt=-f_{d}^{(q)}≤0 and it is zero if and only if 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 nonlocal. This is a consequence of the particular transformation invariance for zero energy of Tsallis' statistics.