  An overarching action principle, the principle of minimal free action, exists
for ergodic Markov chain dynamics. Using this principle and the Detailed
Fluctuation Theorem, we construct a dynamic ensemble theory for non-equilibrium
steady states (NESS) of Markov chains, which is in full analogy with
equilibrium canonical ensemble theory. Concepts such as energy, free energy,
Boltzmann macro-sates, entropy, and thermodynamic limit all have their dynamic
counterparts. For reversible Markov chains, minimization of Boltzmann free
action yields thermal equilibrium states, and hence provide a dynamic
justification of the principle of minimal free energy. For irreversible Markov
chains, minimization of Boltzmann free action selects the stable NESS, and
determines its macroscopic properties, including entropy production. A
quadratic approximation of free action leads to linear-response theory with
reciprocal relations built-in. Hence, in so much as non-equilibrium phenomena
can be modeled as Markov processes, minimal free action serves as a basic
principle for both equilibrium and non-equilibrium statistical physics.
