Non-Markovian approach for anomalous diffusion with infinite memory

The influence of long memory on anomalous diffusion processes is analyzed. We assume that between two successive jumps the moving particle oscillates around an equilibrium position. The number m of oscillations between two jumps is a non-Markovian random variable with infinite memory. The system remembers its whole previous history; all oscillations which have occured in the past have the same probability β of generating a new oscillation in the present. The number mq of oscillations between the qth and the (q + 1)th jumps depends on all previous values m0, m1, …, mq−1 of m. The probability gM(m) that the M = m0 + … + mq+1 previous oscillations generate m oscilations at the qth step is given by a negative binomial gM(m) = βm(1 − β)M(m + M − 1)![m!(M − 1)!]; as a result the total number of oscillations n = M + m increases explosively from step to step and as the process goes on the rate of diffusion is getting smaller and smaller. For a translationally invariant and symmetric diffusion process the asymptotic behavior of the probability density p(r|t) of the position of the moving particle at time t is given by a Gaussian law with a dispersion increasing logarithmically in time; p(r|t) ∼ {[N(-ln(1- β)]2π〈r20〉 ln(vt)]}N2exp[-r2N[-ln(1-β)][2〈r20〉 ln(vy)]], 〈r2(t)〉 ∼ 〈r20〉 ln(vt)[-ln(1-β)]at t → ∞, where 〈r2(t)〉 and 〈r20〉 are the dispersion of the displacement vector at time t and for one jump, respectively, N is the space dimension and v is the frequency of an oscillation.