FLUCTUATING POISSONIAN CLOCKS, FRACTAL RANDOM-PROCESSES AND DYNAMICALPORTER-THOMAS DISTRIBUTIONS - APPLICATIONS TO EVOLUTIONARY MOLECULAR-BIOLOGY, ENHANCED DIFFUSION AND DYNAMICAL RELAXATION

The influence of the memory effects on the Poissonian clocks with fluc tuating counting rates is investigated by using the technique of chara cteristic functionals. A general approach for computing all cumulants of the number of counts is suggested based on an analogy with the theo ry of rate processes with dynamical disorder. The large time behavior of the cumulants is investigated for stationary random processes with short and long memory, respectively. For short memory in the long run all cumulants increase linearly in time and the averaged stochastic pr ocess describing the statistics of the number of counts, although gene rally non-Poissonian, is non-intermittent and can be used as a clock. For finite long memory described by a stationary fractal random proces s, even though the cumulants of the number of events increase faster t han linearly in time, the fluctuations are still non-intermittent and the averaged random process is also a clock. For infinite memory, howe ver, the fluctuations of the number of counts are intermittent and the averaged random process is not a clock any more. An alternative stoch astic approach is developed based on the use of a dynamical analogue o f the Porter-Thomas formula; the results are consistent with the first version of the theory. Three applications of the general theory are p resented The first application is related to the connection between th e Kimura's neutral theory of molecular evolution and the Gillespie's e pisodic clock for the rate of amino acid substitutions through the evo lutionary process. If the fluctuations of the rate of substitution hav e short memory or long finite memory then Gillespie's episodic clock i s consistent with Kimura's theory. Only the infinite memory is not con sistent with the neutral hypothesis. The second application is the stu dy of a hopping mechanism for enhanced diffusion A biased random walk is investigated by assuming that the distribution of the number of jum ps is given by a Poissonian process with a fluctuating counting rate. If the fluctuations of the counting rate have short memory then the re sulting biased diffusion is normal and obeys Einstein's linear equatio n for the mean square displacement of the moving particle. For long me mory the mean square displacement of the moving particle increases fas ter than linearly in time and the diffusion is enhanced. An alternativ e approach for a random walk in the velocity space is developed In thi s case the diffusion process is even more efficient than for a random walk in the real space. The third application is the study of Porter-T homas relaxation for systems with dynamical disorder. It is shown that for small and moderately long times the relaxation function obeys a s caling law of the negative power law type followed by a fast decaying exponential tail which is determined by the fluctuation dynamics. This type of relaxation behavior is of interest both for nuclear and molec ular physics and corresponds to a non-ideal statistical fractal.