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Relaxation Phenomena and Stability of Probability Densities.

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Interim rept. Nov-Dec 84,

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A characteristic function whose positive time behavior is proportional to a step response function is constructed in such a way that all its derivatives at t0 are finite it has the usual exponential decay behavior for intermediate times it satisfies the Paley-Wiener bound for long times. The constructed characteristic function CCF is piecewise continuous with behavior determined by different exponentials of a monomial function of t, termed monomial exponentials, on appropriate segments of time. Continuity conditions at joining points provide relations among the tau sub k so only one tau sub k is an independent parameter. The occurrence of tau sub k well within a particular segment in positive time determines the monomial exponential that dominates the behavior of CCF and the behavior is then called k-dominant. The k-dominance property is discussed for the probability density corresponding to CCF. A formalism is developed in which the probability density for summand variable in omega-space maintains k-dominant behavior for its corresponding characteristic function. The property of k-dominant stability for probability densities is thereby introduced. At this point the identification of the positive t portion of as a step response function is used to make a comparison with a model of relaxation in complex systems which other have called the Ngai model. The latter involves the introduction of interactions that lead to a modification of a constant decay rate for a linear exponential to a time-dependence one appropriate for fractional exponential behavior. Keywords include Relaxation, Fractional exponential, Complex systems.

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  • Statistics and Probability

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