  We briefly review the recently developed, Markov process based isothermal
chemical thermodynamics for nonlinear, driven mesoscopic kinetic systems. Both
the instantaneous Shannon entropy {\boldmath $S[p_{\alpha}(t)]$} and relative
entropy {\boldmath $F[p_{\alpha}(t)]$}, defined based on probability
distribution {\boldmath $\{p_{\alpha}(t)\}$}, play prominent roles. The theory
is general; and as a special case when a chemical reaction system is situated
in an equilibrium environment, it agrees perfectly with Gibbsian chemical
thermodynamics: {\boldmath $k_BS$} and {\boldmath $k_BTF$} become thermodynamic
entropy and free energy, respectively. We apply this theory to a fully
reversible autocatalytic reaction kinetics, represented by a
Delbr\"{u}ck-Gillespie process, in a chemostatic nonequilibrium environment.
The open, driven chemical system serves as an archetype for biochemical
self-replication. The significance of {\em thermodynamically consistent}
kinetic coarse-graining is emphasized. In a kinetic system where death of a
biological organism is treated as the reversal of its birth, the meaning of
mathematically emergent "dissipation", which is not related to the heat
measured in terms of {\boldmath $k_BT$}, remains to be further investigated.
