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...

Under certain circumstances, the time behavior of a random walk is modulated by logarithmic periodic oscillations. The goal of this paper is to present a simple and pedagogical explanation of the origin of this modulation for diffusion on a substrate with two properties: self-similarity and finite ramification order. On these media, the time dependence of the mean-square displacement shows log-periodic modulations around a leading power law, which can be understood on the base of a hierarchical set of diffusion constants. Both the random walk exponent and the period of oscillations are analytically obtained for a pair of examples, one fractal, the other non-fractal, and confirmed by Monte Carlo simulations.

In this work, we report results of extensive computer simulations regarding the phase behavior of a core-softened system. By using structural and thermodynamic descriptors, as well as self-diffusion coefficients, we provide a comprehensive view of the rich phase behavior displayed by the particular instance of the model studied in here. Our calculations agree with previously published results focused on a smaller region in the temperature-density parameter space (Dudalov et al 2014 Soft Matter 10 4966). In this work, we explore a broader region in this parameter space, and uncover interesting fluid phases with low-symmetry local order, that were not reported by previous works. Solid phases were also found, and have been previously characterized in detail by (Kryuchkov et al 2018 Soft Matter 14 2152). Our results support previously reported findings, and provide new physical insights regarding the emergence of order as disordered phases transform into solids by providing radial distribution function maps and specific heat data. Our results are summarized in terms of a phase diagram.