  The seek for a new universal formulation for describing various
non-equilibrium processes is a central task of modern non-equilibrium
thermodynamics. In this paper, a novel steady-state thermodynamic formalism was
established for general Markov processes described by the Chapman-Kolmogorov
equation. Furthermore, corresponding formalisms of steady-state thermodynamics
for master equation and Fokker-Planck equation could be rigorously derived in
mathematics. To be concrete, we proved that: 1) in the limit of continuous
time, the steady-state thermodynamic formalism for the Chapman-Kolmogorov
equation fully agrees with that for the master equation; 2) a similar
one-to-one correspondence could be established rigorously between the master
equation and Fokker-Planck equation in the limit of large system size; 3) when
a Markov process is restrained to one-step jump, the steady-state thermodynamic
formalism for the Fokker-Planck equation with discrete state variables also
goes to that for master equations, as the discretization step gets smaller and
smaller. Our analysis indicated that, with respect to the steady state, general
Markov processes admit a unified and self-consistent non-equilibrium
thermodynamic formulation, regardless of underlying detailed models.
