Linear Free Energy Relations and Reversible Stretched Exponential Kinetics in Systems with Static or Dynamical Disorder

Abstract

Stretched exponential relaxation is the result of the existence of a large number of relaxation channels, any of them having a very small probability of being open. It is shown that the stretched exponential kinetics obeys a type of linear free energy relation. The configuration entropy generated by the random distribution of channels is a linear function of the activation energy of the channel with the slowest relaxation rate and highest energy barrier. This property of stretched exponential relaxation is used for studying the multichannel first-order relaxation kinetics of reversible processes. By combination of the linear free energy relationship with the principle of detailed balance, a generalized kinetic law of the stretched exponential type is derived, which provides a theoretical justification for its prior use in the literature for fitting experimental data. The theory is extended to reversible processes with dynamical disorder. In this case there is no simple analogue of the free energy relationship suggested for systems with static disorder; however, stretched exponential kinetics can be investigated by using a stochastic Liouville equation. It is shown that for a process with dynamical disorder it is possible that in the long time limit the system evolves toward a nonequilibrium frozen state rather than toward thermodynamic equilibrium. We also study the concentration fluctuations for reversible chemical processes in systems with static or dynamical disorder. A set of fluctuation−dissipation relations is derived for the factorial moments of the number of molecules, and it is shown that for both types of disorder the composition fluctuations are intermittent. For the global characterization of the average kinetic behavior of reversible processes occurring in disordered systems we introduce an average lifetime distribution of the transient regime and an effective rate coefficient. The analytic properties of these two functions are investigated for systems with both static and dynamical disorder. Finally, the theory is extended to the case of one-channel thermally activated processes with random energy barriers. We emphasize that our theoretical approach, unlike other theories of stretched exponential relaxation, does not make use of the steepest descent approximation for computing the average kinetic curves: our results are exact in a limit of the thermodynamic type, for which the total number of relaxation channels tends to infinity and the probability that a relaxation channel is open tends to zero, with the constraint that the average number of open channels is kept constant.