Thermodynamics and Geometry of Reversible and Irreversible Markov Processes

Master equation with microscopic reversibility ($q_{ij}\neq 0$ iff $q_{ji}\neq 0$) has a {\em thermodynamic superstructure} in terms of two state functions $S$, entropy, and $F$, free energy: It is discovered recently that entropy production rate $e_p=-dF/dt+Q_{hk}$ with both $-dF/dt=f_d, Q_{hk} \ge 0$. The free energy dissipation $f_d\ge 0$ reflects irreversibility in spontaneous self-organization; house-keeping heat $Q_{hk}\ge 0$ reveals broken time-symmetry in open system driven away from equilibrium. In a Riemannian geometric space, the master equation is a geodesic flow when $Q_{hk}=0$; here we show that the $e_p$ decomposition is orthogonal: $e_p$, $f_d$, $Q_{hk}$ forms a pythagorean triples. Gradient flow means {\em maximum dissipation principle} outside Onsager's regime. The presence of $Q_{hk}$ makses gradient flow no longer generally true. Thermodynamics of stochastic physics requires a new geometric perspective.