Linear free energy relations and reversible stretched exponential kineticsin systems with static car dynamical disorder

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 stud ying the multichannel first-order relaxation kinetics of reversible process es. By combination of the linear free energy relationship with the principl e of detailed balance, a generalized kinetic law of the stretched exponenti al type is derived, which provides a theoretical justification for its prio r use in the literature for fitting experimental data. The theory is extend ed to reversible processes with dynamical disorder. In this case there is n o simple analogue of the free energy relationship suggested for systems wit h static disorder; however, stretched exponential kinetics can be investiga ted by using a stochastic Liouville equation. It is shown that for a proces s with dynamical disorder it is possible that in the long time limit the sy stem evolves toward a nonequilibrium frozen state rather than toward thermo dynamic equilibrium. We also study the concentration fluctuations for rever sible chemical processes in systems wi th static or dynamic al disorder. A set of fluctuation-dissipation relations is derived for the factorial momen ts of the number of molecules, and it is shown that for both types of disor der the composition fluctuations are intermittent, For the global character ization of the average kinetic behavior of reversible processes occurring i n disordered systems we introduce an average Lifetime distribution of the t ransient regime and an effective rate coefficient. The analytic properties of these two functions are investigated for systems with both static and dy namical disorder. Finally, the theory is extended to the case of one-channe l thermally activated processes with random energy barriers. We emphasize t hat our theoretical approach, unlike other theories of stretched exponentia l relaxation, does not make use of the steepest descent approximation for c omputing the average kinetic curves: our results are exact in a limit of th e thermodynamic type, for which the total number of relaxation channels ten ds 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 ke pt constant.