Nonlinear Stochastic Dynamics of Complex Systems, III: Nonequilibrium Thermodynamics of Self-Replication Kinetics

We briefly review the recently developed Markov process-based isothermal chemical thermodynamics for nonlinear driven mesoscopic kinetic systems. Both the instantaneous Shannon entropy
$S[\,p_{\alpha }(t)]$
and relative entropy
$F[\,p_{\alpha }(t)]$
, defined based on probability distribution
$\{\,p_{\alpha }(t);\alpha \in \boldsymbol {\mathscr {S}}\}$
over a state space
$\mathscr {S}$
, 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:
$k_{B}S$
and
$k_{B}TF$
become thermodynamic entropy and free energy, respectively. We apply this theory to a fully reversible autocatalytic reaction kinetics, represented by a Delbrück–Gillespie process, in a chemostatic nonequilibrium environment. The open driven chemical system serves as an archetype for biochemical self-replication and provides insight into a recent theory of England. The significance of
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
$k_{B}T$
, remains to be further investigated.