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)}= summation operator_{i=1}^{N}(p_{i}-p_{i}^{q})/[q(q-1)] and Shiino's generalized free energy F^{(q)}=[ summation operator_{i=1}^{N}p_{i}(p_{i}/pi_{i})^{q-1}-1]/[q(q-1)], where pi_{i} is the stationary distribution. dF^{(q)}/dt=-f_{d}^{(q)}