A GAUSSIAN, NON-MARKOVIAN APPROACH TO LONG MEMORY WITH APPLICATION TO A STOCHASTIC THEORY OF LINE SHAPE

Abstract

A new approach to long memory effects is suggested. The main assumption is that the regression of fluctuations occurs via a clustering mechanism. The regression of a scalar variable takes place through a succession of linear decay processes, resulting in a long time tail of the autocorrelation function. The number of decay events is a random variable obeying a self-similar probability distribution whose fractal exponent determines the long time behavior of fluctuation regression. The overall fluctuation dynamics is described by a stationary Gaussian and non-Markovian random process. The theory is applied to the stochastic theory of line shape. The relaxation function ϕ(t) can be exactly evaluated. We show that the memory effects lead to a narrowing of the relaxation function for large time. As the time t tends to infinity, ϕ(t) may be approximated by a ‘contracted’ exponential ϕ(t) ~ exp(-const. t2−H) where 1≥H>0 is the fractal exponent describing the clustering process.