https://doi.org/10.1140/epjb/e2012-30378-5
Regular Article
Superdiffusion in a non-Markovian random walk model with a Gaussian memory profile
1
Departamento de Física Teórica e Experimental, Universidade
Federal do Rio Grande do Norte, 59078-900
Natal, RN, Brazil
2
Departamento de Física, Universidade Federal de
Pernambuco, 50670-901
Recife, PE, Brazil
3
Departamento de Física e Química, FCFRP, Universidade de São
Paulo, 14040-903
Ribeirão Preto, SP, Brazil
4
Instituto de Física, Universidade Federal de
Alagoas, 57072-970
Maceió, AL, Brazil
a e-mail: gislene@dfte.ufrn.br
Received:
11
May
2012
Received in final form:
27
July
2012
Published online:
12
September
2012
Most superdiffusive Non-Markovian random walk models assume that correlations are maintained at all time scales, e.g., fractional Brownian motion, Lévy walks, the Elephant walk and Alzheimer walk models. In the latter two models the random walker can always “remember” the initial times near t = 0. Assuming jump size distributions with finite variance, the question naturally arises: is superdiffusion possible if the walker is unable to recall the initial times? We give a conclusive answer to this general question, by studying a non-Markovian model in which the walker’s memory of the past is weighted by a Gaussian centered at time t/2, at which time the walker had one half the present age, and with a standard deviation σt which grows linearly as the walker ages. For large widths we find that the model behaves similarly to the Elephant model, but for small widths this Gaussian memory profile model behaves like the Alzheimer walk model. We also report that the phenomenon of amnestically induced persistence, known to occur in the Alzheimer walk model, arises in the Gaussian memory profile model. We conclude that memory of the initial times is not a necessary condition for generating (log-periodic) superdiffusion. We show that the phenomenon of amnestically induced persistence extends to the case of a Gaussian memory profile.
Key words: Statistical and Nonlinear Physics
© EDP Sciences, Società Italiana di Fisica and Springer-Verlag, 2012
