Random walk on fractals: numerical studies in two dimensions

Heterogeneous Mean First-Passage Time Scaling in Fractal Media

The mean first passage time (MFPT) of random walks is a key quantity characterizing dynamic processes on disordered media. In a random fractal embedded in the Euclidean space, the MFPT is known to obey the power law scaling with the distance between a source and a target site with a universal exponent. We find that the scaling law for the MFPT is not determined solely by the distance between a source and a target but also by their locations. The role of a site in the first passage processes is quantified by the random walk centrality. It turns out that the site of highest random walk centrality, dubbed as a hub, intervenes in first passage processes. We show that the MFPT from a departure site to a target site is determined by a competition between direct paths and indirect paths detouring via the hub. Consequently, the MFPT displays a crossover scaling between a short distance regime, where direct paths are dominant, and a long distance regime, where indirect paths are dominant. The two regimes are characterized by power laws with different scaling exponents. The crossover scaling behavior is confirmed by extensive numerical calculations of the MFPTs on the critical percolation cluster in two dimensional square lattices.

Subdiffusive continuous-time random walks with stochastic resetting

We analyze two models of subdiffusion with stochastic resetting. Each of them consists of two parts: subdiffusion based on the continuous-time random walk scheme and independent resetting events generated uniformly in time according to the Poisson point process. In the first model the whole process is reset to the initial state, whereas in the second model only the position is subject to resets. The distinction between these two models arises from the non-Markovian character of the subdiffusive process. We derive exact expressions for the two lowest moments of the full propagator, stationary distributions, and first hitting time statistics. We also show, with an example of a constant drift, how these models can be generalized to include external forces. Possible applications to data analysis and modeling of biological systems are also discussed.

Random walks with fractally correlated traps: Stretched exponential and power-law survival kinetics

We consider the survival probability f(t) of a random walk with a constant hopping rate w on a host lattice of fractal dimension d and spectral dimension d_{s}≤2, with spatially correlated traps. The traps form a sublattice with fractal dimension d_{a}<d and are characterized by the absorption rate w_{a} which may be finite (imperfect traps) or infinite (perfect traps). Initial coordinates are chosen randomly at or within a fixed distance of a trap. For weakly absorbing traps (w_{a}≪w), we find that f(t) can be closely approximated by a stretched exponential function over the initial stage of relaxation, with stretching exponent α=1-(d-d_{a})/d_{w}, where d_{w} is the random walk dimension of the host lattice. At the end of this initial stage there occurs a crossover to power-law kinetics f(t)∼t^{-α} with the same exponent α as for the stretched exponential regime. For strong absorption w_{a}≳w, including the limit of perfect traps w_{a}→∞, the stretched exponential regime is absent and the decay of f(t) follows, after a short transient, the aforementioned power law for all times.

Geometry controlled anomalous diffusion in random fractal geometries: looking beyond the infinite cluster

We investigate the ergodic properties of a random walker performing (anomalous) diffusion on a random fractal geometry. Extensive Monte Carlo simulations of the motion of tracer particles on an ensemble of realisations of percolation clusters are performed for a wide range of percolation densities. Single trajectories of the tracer motion are analysed to quantify the time averaged mean squared displacement (MSD) and to compare this with the ensemble averaged MSD of the particle motion. Other complementary physical observables associated with ergodicity are studied, as well. It turns out that the time averaged MSD of individual realisations exhibits non-vanishing fluctuations even in the limit of very long observation times as the percolation density approaches the critical value. This apparent non-ergodic behaviour concurs with the ergodic behaviour on the ensemble averaged level. We demonstrate how the non-vanishing fluctuations in single particle trajectories are analytically expressed in terms of the fractal dimension and the cluster size distribution of the random geometry, thus being of purely geometrical origin. Moreover, we reveal that the convergence scaling law to ergodicity, which is known to be inversely proportional to the observation time T for ergodic diffusion processes, follows a power-law ∼T(-h) with h < 1 due to the fractal structure of the accessible space. These results provide useful measures for differentiating the subdiffusion on random fractals from an otherwise closely related process, namely, fractional Brownian motion. Implications of our results on the analysis of single particle tracking experiments are provided.

Random walks on Sierpinski gaskets of different dimensions

We study random walks (RWs) on classical and dual Sierpinski gaskets (SG and DSG), naturally embedded in d-dimensional Euclidian spaces (ESs). For large d the spectral dimension d(s) approaches 2, the marginal RW dimension. In contrast to RW over two-dimensional ES, RWs over SG and DSG show a very rich behavior. First, the time discrete scale invariance leads to logarithmic-periodic (log-periodic) oscillations in the RW properties monitored, which increase in amplitude with d. Second, the asymptotic approach to the theoretically predicted RW power laws is significantly altered depending on d and on the variant of the fractal (SG or DSG) under study. In addition, we discuss the suitability of standard RW properties to determine d(s), a question of great practical relevance.

Global first-passage times of fractal lattices

The global first passage time density of a network is the probability that a random walker released at a random site arrives at an absorbing trap at time T . We find simple expressions for the mean global first passage time <T> for five fractals: the d-dimensional Sierpinski gasket, T fractal, hierarchical percolation model, Mandelbrot-Given curve, and a deterministic tree. We also find an exact expression for the second moment <T(2)> and show that the variance of the first passage time, Var(T) , scales with the number of nodes within the fractal N such that Var(T) approximately N(4/d[over]), where d[over] is the spectral dimension.

Probing microscopic origins of confined subdiffusion by first-passage observables

Subdiffusive motion of tracer particles in complex crowded environments, such as biological cells, has been shown to be widespread. This deviation from Brownian motion is usually characterized by a sublinear time dependence of the mean square displacement (MSD). However, subdiffusive behavior can stem from different microscopic scenarios that cannot be identified solely by the MSD data. In this article we present a theoretical framework that permits the analytical calculation of first-passage observables (mean first-passage times, splitting probabilities, and occupation times distributions) in disordered media in any dimensions. This analysis is applied to two representative microscopic models of subdiffusion: continuous-time random walks with heavy tailed waiting times and diffusion on fractals. Our results show that first-passage observables provide tools to unambiguously discriminate between the two possible microscopic scenarios of subdiffusion. Moreover, we suggest experiments based on first-passage observables that could help in determining the origin of subdiffusion in complex media, such as living cells, and discuss the implications of anomalous transport to reaction kinetics in cells.

Analytic expression for the mean time to absorption for a random walker on the Sierpinski gasket

The exact analytic expression for the mean time to absorption (or mean walk length) for a particle performing a random walk on a finite Sierpinski gasket with a trap at one vertex is found to be T((n))=[3(n)5(n+1)+4(5(n))-3(n)]/(3(n+1)+1) where n denotes the generation index of the gasket, and the mean is over a set of starting points of the walk distributed uniformly over all the other sites of the gasket. In terms of the number N(n) of sites on the gasket and the spectral dimension d of the gasket, the precise asymptotic behavior for large N(n) is T((n))-->1/3(2N(n))(2/d)-N1.464. This serves as a partial check on our result, as it is (a) intermediate between the known results T-N2 (d=1) and T-N ln N (d=2) for random walks on d-dimensional Euclidean lattices and (b) consistent with the known result for the asymptotic behavior of the mean number of distinct sites visited in a random walk on a fractal lattice.

Territory covered by N random walkers on fractal media: the Sierpinski gasket and the percolation aggregate

We address the problem of evaluating the number S(N)(t) of distinct sites visited up to time t by N noninteracting random walkers all starting from the same origin in fractal media. For a wide class of fractals (of which the percolation cluster at criticality and the Sierpinski gasket are typical examples) we propose, for large N and after the short-time compact regime, an asymptotic series for S(N)(t) analogous to that found for Euclidean media: S(N)(t) approximately S(N)(t)(1-Delta). Here S(N)(t) is the number of sites (volume) inside a hypersphere of radius L[ln(N)/c]1/v where L is the root-mean-square chemical displacement of a single random walker, and v and c determine how fast 1-Gamma(t)(l) (the probability that a given site at chemical distance l from the origin is visited by a single random walker by time t) decays for large values of l/L: 1-Gamma(t)(l) approximately exp[-c(l/L)(v)]. For the fractals considered in this paper, v=d(l)w/((d(l)w)-1), d(l)w being the chemical-diffusion exponent. The corrective term Delta is expressed as a series in ln(-n)(N)ln(m) ln(N) (with n> or =1 and 0< or =m< or =n), which is given explicitly up to n=2. This corrective term contributes substantially to the final value of S(N)(t) even for relatively large values of N.

Time course of reactions controlled and gated by intramolecular dynamics of proteins: predictions of the model of random walk on fractal lattices

Computer simulations of random walk on the Sierpinski gasket and percolation clusters demonstrate that the short, initial condition-dependent stage of protein involving reactions can dominate the progress of the reaction over the main stage described by the standard kinetics. This phenomenon takes place if the intramolecular conformational transition dynamics modeled by the stochastic process is slow enough and the initial conformational substate of the protein already belongs to the transition state of the reaction. Both conditions are realized in two kinds of experiments: small ligand rebinding to protein after laser flash photolysis and direct recording of single protein channel activity. The model considered suggests simple analytical formulae that can explain the time behavior of the processes observed and its variation with temperature. The initial condition-dependent stage, and not the stage described by the standard kinetics, is expected as responsible for the coupling of component reactions in the complete catalytic cycles and more complex processes of biological free energy transduction.

A synthetic picture of intramolecular dynamics of proteins. Towards a contemporary statistical theory of biochemical processes

An increasing body of experimental evidence indicates the slow character of internal dynamics of native proteins. The important consequence of this is that theories of chemical reactions, used hitherto, appear inadequate for description of most biochemical reactions. Construction of a contemporary, truly advanced statistical theory of biochemical processes will need simple but realistic models of microscopic dynamics of biomolecules. In this review, intended to be a contribution towards this direction, three topics are considered. First, an intentionally simplified picture of dynamics of native proteins which emerges from recent investigations is presented. Fast vibrational modes of motion, of periods varying from 10(-14) to 10(-11) s, are contrasted with purely stochastic conformational transitions. Significant evidence is adduced that the relaxation time spectrum of the latter spreads in the whole range from 10(-11) to 10(5) s or longer, and up to 10(-7) s it is practically quasi-continuous. Next, the essential ideas of the theory of reaction rates based on stochastic models of intramolecular dynamics are outlined. Special attention is paid to reactions involving molecules in the initial conformational substrates confirmed to the transition state, which is realized in actual experimental situations. And finally, the two best experimentally justified classes of models of conformational transition dynamics, symbolically referred to as "protein glass" and "protein machine", are described and applied to the interpretation of a few simple biochemical processes, perhaps the most important result reported is the demonstration of the possibility of predominance of the short initial condition-dependent stage of protein involved reactions over the main stage described by the standard kinetics. This initial stage, and not the latter, is expected to be responsible for the coupling of component reactions in the complete enzymatic cycles as well as more complex processes of biological free energy transduction.