LONG MEMORY, FRACTAL STATISTICS, AND ANDERSON LOCALIZATION FOR CHEMICAL WAVES AND PATTERNS WITH RANDOM PROPAGATION VELOCITIES

An analytic approach is developed for computing the moments of concent ration fields in a spatially inhomogeneous chemical system subject to environmental fluctuations, based on phase linearization. It is shown that the environmental fluctuations lead to Anderson localization. If in the absence of environmental fluctuations the system displays chemi cal waves periodic in space and time, then in the presence of fluctuat ions the waves become localized in time and space. Two limit cases exi st:for homogeneous chemical systems displaying chemical oscillations, the environmental fluctuations lead to damped oscillations, i.e., to t emporal localization, whereas for structured periodic patterns the loc alization occurs only in space. The validity of the suggested approach is tested by investigating the behavior of one-dimensional reaction-c onvection systems subject to time-dependent and space-independent velo city fluctuations. Computations are performed in the case of non-Marko vian Gaussian perturbations of the velocity field. Both analytical and numerical calculations show that the Anderson localization of the con centration patterns is very strong for non-Markovian fluctuations with long memory characterized by correlation functions of the negative po wer-law type. For infinite memory the attenuation factors are Gaussian . For self-similar fractal random processes with long but finite memor y, the localization is less strong and the attenuation factor is given by a compressed exponential and has a shape intermediate between a Ga ussian and an exponential. Finally, for Markovian or independent rando m processes the localization is weak and the attenuation is exponentia l. We suggest an experiment for testing the predicted theoretical resu lts and discuss the possibilities of generalizing the theory for react ion-convection systems with thermal fluctuations and for Levy noise by using the Shlesinger-Hughes renormalization technique.