Rate statistics and thermodynamic analogies for relaxation processes in systems with static disorder: Application to stretched exponential

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Abstract
The paper deals with the relationships between the total rate of a relaxation process occurring in a system with static disorder and the decay rates attached to the different individual reaction channels. It is proven that the models of relaxation constructed on the basis of these two types of rates are equivalent to each other. From an experimentally observed relaxation curve it is possible to evaluate only the density of channels characterized by different relaxation rates and the overall probability distribution of the total relaxation rate. For evaluating the probability density of the individual relaxation rates attached to different channels an approach based on the maximum information entropy principle is suggested. A statistical thermodynamic formalism is developed for the relaxation time of a given channel, i.e., for the reciprocal value of the individual relaxation rate. The probability density of the relaxation time is proportional to the product of the density of channels to an exponentially decreasing function similar to the Boltzmann’s factor in equilibrium statistical mechanics. The theory is applied to the particular case of stretched exponential relaxation for which the density of channels diverges to infinity in the limit of large relaxation times according to a power law. The extremal entropy of the system as well as the moments and the cumulants of the relaxation times and of the relaxation rates are evaluated analytically. The probability of fluctuations can be expressed by a relationship similar to the Greene–Callen generalization of Einstein’s fluctuation formula. In the limit of large rates the density of channels and the probability density of individual rates have the same behavior; both functions have long tails of the negative power law type characterized by the same fractal exponent. For small rates, however, their behavior is different; the probability density tends to zero in the limit of very small rates whereas the density of channels displays an infrared divergence in the same region and tends to infinity. Although in the limit of small rates the density of channels is very large the probability of occurrence of these channels is very small; the compensation between these two opposite factors leads to the self-similar features displayed by the stretched exponential relaxation. The thermodynamic approach is compared with a model calculation for the problem of direct energy transfer in finite systems. The connections between stretched exponential relaxation and the thermal activation of the channels are also investigated. It is shown that stretched exponential relaxation corresponds to a distribution of negative and positive activation energies of the Gompertz-type.