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

Saakian, David B. ; Qian, Hong ;
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    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.


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