FRACTAL TIME, ULTRAMETRIC TOPOLOGY AND FAST RELAXATION

An ultrametric model for very fast relaxation processes is suggested. We assume that a relaxation step is a succession of series-parallel el ementary processes organized in a hierarchical structure formed from b ranches (relaxation channels) grouped in levels. The model is characte rized by the following parameters: the average number [n]0 of channels from the zeroth level of the hierarchy; the probability beta that in a given level a new channel is generated, the frequency omega0 of an e lementary process and the probability of decay beta attached to an ove rall relaxation step. The number of relaxation channels increases expo nentially with the level index; as a result the survival probability o f the model, l(t), decreases much faster than the usual exponential la w: l(t) = {-[n]0p[(1/beta)-1][exp[omega0tbeta/(1-beta)]-1]}. Due to th is double exponential decay not only all positive moments of the lifet ime probability density psi(t) = -partial derivative l(t) lat exist an d are finite, but also all positive moments of the exponential of the lifetime, exp(mut), mu > 0, exist and are finite. The relationships am ong the fast relaxation (1), the exponential (Markovian) relaxation (2 ) and the fractal time (slow) relaxation (3) are clarified: in the suc cession (1) --> (2) --> (3) each relaxation law can be obtained from t he preceding one through a logarithmic transformation of the time vari able. 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 b e obtained from the isoperimetric condition of the preceding law by a logarithmic transformation of time. The fast relaxation and the fracta l time laws play symmetrical roles in comparison with the Markovian re laxation. The fast relaxation corresponds to an ''antifractal'' behavi or.