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A complete mathematical theory of nonequilibrium thermodynamics of stochasticsystems in terms of master equations is presented. As generalizations ofisothermal entropy and free energy, two functions of states play central roles:the Gibbs entropy $S$ and the relative entropy $F$, which are related via thestationary distribution of the stochastic dynamics. $S$ satisfies thefundamental entropy balance equation $dS/dt=e_p-h_d/T$ with entropy productionrate $e_p\ge 0$ and heat dissipation rate $h_d$, while $dF/dt=-f_d\le 0$. Forclosed systems that satisfy detailed balance: $Te_p(t)=f_d(t)$. For open systemone has $Te_p(t)=f_d(t)+Q_{hk}(t)$ where the housekeeping heat $Q_{hk}\ge 0$was first introduced in the phenomenological nonequilibrium steady statethermodynamics. Entropy production $e_p$ consists of free energy dissipationassociated with spontaneous relaxation, $f_d$, and active energy pumping thatsustains the open system $Q_{hk}$. The amount of excess heat involved in therelaxation $Q_{ex}=h_d-Q_{hk} = f_d-T(dS/dt)$.

The Physical Origins of Entropy Production, Free Energy Dissipation and their Mathematical Representations