Non-Markovian approach for anomalous diffusion with infinite memory

Marcel Ovidiu Vlad

Physica A: Statistical Mechanics and its Applications, 1994, vol. 208, issue 2, 167-176

Abstract: 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.

Date: 1994
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:208:y:1994:i:2:p:167-176

DOI: 10.1016/0378-4371(94)00019-0

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