Eur. Phys. J. B 72, 427-433 (2009)
https://doi.org/10.1140/epjb/e2009-00361-6

Sudden onset of log-periodicity and superdiffusion in non-Markovian random walks with amnestically induced persistence: exact results

¹ Instituto de Física, Universidade Federal de Alagoas, 57072-970 Maceió, AL, Brazil
² Departamento de Física e Química, FCFRP, Universidade de São Paulo, Ribeirão Preto, 14040-903 SP, Brazil
³ Consortium of the Americas for Interdisciplinary Science and Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico, 87131, USA

Corresponding author: ^a mflmatrix@gmail.com

Received: 19 March 2009
Revised: 12 August 2009
Published online: 27 October 2009

Abstract

Random walks can undergo transitions from normal diffusion to anomalous diffusion as some relevant parameter varies, for instance the Lévy index in Lévy flights. Here we derive the Fokker-Planck equation for a two-parameter family of non-Markovian random walks with amnestically induced persistence. We investigate two distinct transitions: one order parameter quantifies log-periodicity and discrete scale invariance in the first moment of the propagator, whereas the second order parameter, known as the Hurst exponent, describes the growth of the second moment. We report numerical and analytical results for six critical exponents, which together completely characterize the properties of the transitions. We find that the critical exponents related to the diffusion-superdiffusion transition are identical in the positive feedback and negative feedback branches of the critical line, even though the former leads to classical superdiffusion whereas the latter gives rise to log-periodic superdiffusion.