Fractal time, ultrametric topology and fast relaxation

Abstract

An ultrametric model for very fast relaxation processes is suggested. We assume that a relaxation step is a succession of series- parallel elementary processes organized in a hierarchical structure formed from branches (relaxation channels) grouped in levels. The model is characterized by the following parameters: the average number 〈 n〉 0 of channels from the zeroth level of the hierarchy; the probability β that in a given level a new channel is generated, the frequency ω 0 of an elementary process and the probability of decay p attached to an overall relaxation step. The number of relaxation channels increases exponentially with the level index; as a result the survival probability of the model, l(t), decreases much faster than the usual exponential law: l(t) = exp { - 〈n〉 0p[( 1 β )−1] [ exp[ ω 0tβ (1−β) ]−s1]} . Due to this double exponential decay not only all positive moments of the lifetime probability density ϕ(t) = - ∂l(t) ∂(t) exist and are finite, but also all positive moments of the exponential of the lifetime, exp(μ t), μ> 0, exist and are finite. The relationships among the fast relaxation (1), the exponential (Markovian) relaxation (2) and the fractal time (slow) relaxation (3) are clarified: in the succession (1) → (2) → (3) each relaxation law can be obtained from the preceding one through a logarithmic transformation of the time variable. These three laws can be obtained by maximizing the informational entropy of the lifetime distribution; in the succession (1) → (2) → (3) the isoperimetric condition corresponding to a given law can be obtained from the isoperimetric condition of the preceeding law by a logarithmic transformation of time. The fast relaxation and the fractal time laws play symmetrical roles in comparison with the Markovian relaxation. The fast relaxation corresponds to an “antifractal” behavior.