Log-periodic modulation in one-dimensional random walks

Abstract

Abstract. -We have studied the diffusion of a single particle on a one-dimensional lattice. It is shown that, for a self-similar distribution of hopping rates, the time dependence of the mean-square displacement follows an anomalous power law modulated by logarithmic periodic oscillations. The origin of this modulation is traced to the dependence on the length of the diffusion coefficient. Both the random walk exponent and the period of the modulation are analytically calculated and confirmed by Monte Carlo simulations.Brownian Motion is a well-known phenomenon, and since its theoretical foundations were laid, more than one hundred years ago [1,2], diffusion processes and random walk models have been attracting the attention of researchers. The importance of random walk (RW) resides in the fact that it is the simplest realisation of Brownian Motion, with applications in almost every field of science where stochastic dynamics play a role [3]. It is worth to remark that, even though a random walker evolves according to simple rules, a considerable effort may be needed to solve the dynamic problem in detail and unexpected behaviours may emerge. Thus, for example, our understanding of the mechanisms responsible for anomalous diffusion has been strongly fluenced by the large amount of work devoted to the study of RW in non-Euclidean media, during the last three decades [4][5][6].In the last years, it has been often reported that, sometimes, the time behaviour of a RW is modulated by logarithmic-periodic oscillations. These fluctuations has been rigorously studied by mathematicians on special kinds of graphs. A proof of the fluctuating behaviour of the n-step probabilities for a simple RW on a Sierpiński graph was given in ref. [7] and a generalisation to the broad class of symmetrically self-similar graphs can be found in ref. [8]. Within the physical community, it has been shown that, on Sierpiński gaskets, the mean number of distinct sites visited at time t by N noninteracting random walkers presents an oscillatory behaviour [9] and, more recently, detailed studies of the log-periodic modulations on fractals with finite ramification order, were presented in refs. [10,11].Log-periodic modulations are not restricted to random walks. It is in general believed that they appear because of an inherent self-similarity [12], responsible for a discrete scale invariance (DSI) [13]. Examples of these oscillations have been detected in earthquakes [14,15], escape probabilities in chaotic maps close to crisis [16], biased diffusion [17,18], kinetic and dynamic processes on random quenched and fractal media [19][20][21][22], and stock markets near a financial crash [23][24][25][26].In this work we analyse a minimal model of RW, which results in log-periodic modulations of some observables. The main objective is to investigate the underlying physics of the oscillatory behaviour mentioned above. Sometimes, physical phenomena can be more easily grasped with the help of simple models. Thus, the present study may be useful to determi...

Log-periodic oscillations for diffusion on self-similar finitely ramified structures

“…In this paper, we revisit this problem, with the purpose of explaining the origin of the behavior described above in a simple * Fellow of Consejo Nacional de Investigaciones Científicas y Téc-nicas (CONICET) † Research Member of CONICET ‡ Electronic address: iguain@mdp.edu.ar and comprehensible way. The work is a natural continuation of a previous one [31], where we studied log-periodic modulation in one-dimensional RW.…”

Robustness of the non-Markovian Alzheimer walk under stochastic perturbation

“…The properties of the phases include log-periodic oscillations that appear for small sizes of the long-range memory. Log-periodic oscillations in RW have been reported to appear elsewhere (see e.g., [21][22][23]). We also show that the size of the region of the phase diagram with superdiffusion is controlled by the memoryless noise.…”

Universal behavior for single-file diffusion on a disordered fractal

“…Single-file diffusion on a fractal has been analyzed in Ref. [26], for a set of hard-core interacting particles moving on a one-dimensional lattice with a self-similar distribution of hopping-rates [12]. In this problem, after a finite time, the tagged-particle MSD shows global subdiffusive behavior (∼ t ν , with ν < 1/2) modulated by logarithmic-periodic oscillations.…”