RATE STATISTICS AND THERMODYNAMIC ANALOGIES FOR RELAXATION PROCESSES IN SYSTEMS WITH STATIC DISORDER - APPLICATION TO STRETCHED EXPONENTIAL

The paper deals with the relationships between the total rate of a rel axation process occurring in a system with static disorder and the dec ay rates attached to the different individual reaction channels. It is proven that the models of relaxation constructed on the basis of thes e two types of rates are equivalent to each other. From an experimenta lly observed relaxation curve it is possible to evaluate only the dens ity of channels characterized by different relaxation rates and the ov erall probability distribution of the total relaxation rate. For evalu ating the probability density of the individual relaxation rates attac hed 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 th e reciprocal value of the individual relaxation rate. The probability density of the relaxation time is proportional to the product of the d ensity of channels to an exponentially decreasing function similar to the Boltzmann's factor in equilibrium statistical mechanics. The theor y is applied to the particular case of stretched exponential relaxatio n for which the density of channels diverges to infinity in the limit of large relaxation times according to a power law. The extremal entro py of the system as well as the moments and the cumulants of the relax ation times and of the relaxation rates are evaluated analytically. Th e probability of fluctuations can be expressed by a relationship simil ar to the Greene-Callen generalization of Einstein's fluctuation formu la. In the limit of large rates the density of channels and the probab ility density of individual rates have the same behavior; both functio ns have long tails of the negative power law type characterized by the same fractal exponent. For small rates, however, their behavior is di fferent; the probability density tends to zero in the limit of very sm all rates whereas the density of channels displays an infrared diverge nce 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 t hese two opposite factors leads to the self-similar features displayed by the stretched exponential relaxation. The thermodynamic approach i s compared with a model calculation for the problem of direct energy t ransfer in finite systems. The connections between stretched exponenti al relaxation and the thermal activation of the channels are also inve stigated. It is shown that stretched exponential relaxation correspond s to a distribution of negative and positive activation energies of th e Gompertz-type. (C) 1997 American Institute of Physics.