Random walks (RWs) and related stochastic techniques have become ubiquitous tools in many areas of physics recently. Fractals are no exception. Random walks on fractals have an added interest: random walk trails (e.g. sample paths of Brownian motion) are themselves fractal in general, and interesting kinds of behavior emerge when they occur on fractal structures, in the form of scaling laws. In the case of random Factals, one has a "random walk on a random walk". This is an instance of a so-called "random" random walk, in the study of which considerable progress has recently been made. © 1995.

V. Balakrishnan

Fractals

Markov processes

Percolation (solid state)

Probability

Random processes

FRACTAL STRUCTURE

Random Walks

Diffusion

IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

Random walks on fractals

Materials Science and Engineering B

We consider a mortal random walker on a family of hierarchical graphs in the presence of some trap sites. The configuration comprising the graph, the starting point of the walk, and the locations of the trap sites is taken to be exactly self-similar as one goes from one generation of the family to the next. Under these circumstances, the total probability that the walker hits a trap is determined exactly as a function of the single-step survival probability q of the mortal walker. On the nth generation graph of the family, this probability is shown to be given by the nth iterate of a certain scaling function or map q→f(q). The properties of the map then determine, in each case, the behavior of the trapping probability, the mean time to trapping, the temporal scaling factor governing the random walk dimension on the graph, and other related properties. The formalism is illustrated for the cases of a linear hierarchical lattice and the Sierpinski graphs in two and three Euclidean dimensions. We find an effective reduction of the random walk dimensionality due to the ballistic behavior of the surviving particles induced by the mortality constraint. The relevance of this finding for experiments involving travel times of particles in diffusion-decay systems is discussed. © 2019 American Physical Society.

Fulltext

Physical Review E

First-passage properties of mortal random walks: Ballistic behavior, effective reduction of dimensionality, and scaling functions for hierarchical graphs

We present exact, analytic results for the mean time to trapping of a random walker on the class of deterministic Sierpinski graphs embedded in d≥2 Euclidean dimensions, when both nearest-neighbor (NN) and next-nearest-neighbor (NNN) jumps are included. Mean first-passage times are shown to be modified significantly as a consequence of the fact that NNN transitions connect fractals of two consecutive generations. © 2013 American Physical Society.

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Analytic expression for the mean time to absorption for a random walker on the Sierpinski fractal. III. the effect of non-nearest-neighbor jumps

The formula for the mean length of a random walk on the Sierpinski tower was described. An unbiased random walk on a finite, nth generation Sierpinski gasket, in the presence of a trap at one vertex, was considered. The generalization of the formula for the case of a tower embedded in an arbitrary number of Euclidean dimensions was also found.

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Exact formula for the mean length of a random walk on the Sierpinski tower

In view of the interest in the occurrence of anomalous diffusion (〈r2(t)〉 ∼ t2H, 0 &lt; H &lt; 1 2) in several physical circumstances, we study anomalous diffusion per se in terms of exactly solvable one-dimensional models. The basic idea is to exploit the fact that temporal correlations lead directly to anomalous diffusion, and provide solvable analogues of more realistic physical situations. We first derive a general equation for a deterministic trajectory xε(t) that comprehensively characterizes the diffusive motion, by finding the ε-quantiles of the time-dependent probability distribution. The class of all diffusion processes (or, equivalently, symmetric random walks) for which xε(t) ∼ t1 2, and, subsequently, xε(t) ∼ tH, is identified. Explicit solutions are presented for families of such processes. Considering random walks whose step sequences in time are governed by renewal processes, and proceeding to the continuum limit, a true generalization of Brownian motion (the latter corresponds to the limiting value H = 1 2) is obtained explicitly: 〈x2(t)〉 ∼ t2H; the diffusive spread of the initial condition is given by xε(t) ∼ tH; and the first passage time from the origin to the point x has a stable Lévy distribution with an exponent equal to H. © 1985.

Physica A: Statistical Mechanics and its Applications

Anomalous diffusion in one dimension

Benezit Dictionary of Artists

Broeck, Crispin van den

Berichte der Bunsengesellschaft für physikalische Chemie

First Passage Times and Transport in Systems with Disorder

Physics Reports

Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications

Physical Review A

Renormalization of first-passage times for random walks on deterministic fractals

Journal	Materials Science and Engineering B
ISSN	09215107
Open Access	No