Random spin models and chemical kinetics

Abstract

Lyapunov exponents constitute a class of parameters which describe the asymptotic behavior of a large class of dynamical processes in chemistry and physics. This paper gives a variational characterization of the largest Lyapunov exponent for a class of models of chemical kinetics described by products of random nonnegative matrices. We show that for this class of models the largest Lyapunov exponent satisfies an extremal principle formally identical to the minimization of the quenched free energy in random spin models. This extremal principle, which yields a computable expression for the Lyapunov exponent, implies that fluctuations in the Lyapunov exponent, due to a certain class of perturbations in the matrix elements, are determined by a macroscopic parameter which is the analog of the mean energy in random spin systems. These results characterize a class of random models in chemical kinetics that are thermodynamically stable in the sense that they possess an asymptotic limit in which analogs of the laws of equilibrium thermodynamics hold.