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) = Sigma(N)(i=1) (pi - p(i)(q))/[q(q - 1)] and Shiino's generalized free energy F(q) = [Sigma(N)(i=1) pi(p(i)/pi(i))(q-1) - 1]/[q(q - 1)], where pi(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.