Relaxation phenomena and stability of probability densities

A characteristics function whose positive time behavior is proportional to a step response function is constructed in such a way that: all its derivatives at t = 0 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 Cπk(t|0) is piecewise continuous with behavior determined by different exponentials of a monomial function of t, namely −(\t\/τk)k, termed monomial exponentials, on appropriate segments of time. Continuity conditions at joining points provide relations among the τk so only one τk is an independent parameter. The occurrence of τk well within a particular segment in (positive) time determines the monomial exponential that dominates the behavior of Cτk(t|0), and the behavior is then called k-dominant. The k-dominance property is discussed for the probability density Pτk(ω) corresponding to Cτk(t|0). A formalism is developed in which the probability density for a summand variable in ω-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 Cτk (t|0) as a step response function is used to make a comparison which a relaxation model in complex system which others 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-dependent one appropriate for fractional expotential behavior. Ngai's predicted relation between the two respective relaxation time parameters corresponds here to a continuity condition for Cτk(t|0). k-dominance and, by implication, k-dominant stability are compatible with the Ngai model. Also compatible with the Ngai model and the concept of k-dominant stability is a cross-over from one-dominant to α-dominant behavior under a change of experimental conditions that is actually observed for selected systems.