  Motivated by a recent work on the metabolism of carbohydrates in bacteria, we
study the kinetics and thermodynamics of two classic models for reversible
polymerization, one preserving the total polymer concentration and the other
one not. The chemical kinetics is described by rate equations following the
mass-action law. We consider a closed system and nonequilibrium initial
conditions and show that the system dynamically evolves towards equilibrium
where detailed balance is satisfied. The entropy production during this process
can be expressed as the time derivative of a Lyapunov function. When the
solvent is not included in the description and the dynamics conserves the total
concentration of polymer, the Lyapunov function can be expressed as a
Kullback-Leibler divergence between the nonequilibrium and the equilibrium
polymer length distribution. The same result holds true when the solvent is
explicitly included in the description and the solution is assumed dilute,
whether or not the total polymer concentration is conserved. Furthermore, in
this case a consistent nonequilibrium thermodynamic formulation can be
established and the out-of-equilibrium thermodynamic enthalpy, entropy and free
energy can be identified. Such a framework is useful in complementing standard
kinetics studies with the dynamical evolution of thermodynamic quantities
during polymerization.
