Two-state random walk model of diffusion. 2. Oscillatory diffusion

Continuing our study of interrupted diffusion, we consider the problem of a particle executing a random walk interspersed with localized oscillations during its halts (e.g., at lattice sites). Earlier approaches proceedvia approximation schemes for the solution of the Fokker-Planck equation for diffusion in a periodic potential. In contrast, we visualize a two-state random walk in velocity space with the particle alternating between a state of flight and one of localized oscillation. Using simple, physically plausible inputs for the primary quantities characterising the random walk, we employ the powerful continuous-time random walk formalism to derive convenient and tractable closed-form expressions for all the objects of interest: the velocity autocorrelation, generalized diffusion constant, dynamic mobility, mean square displacement, dynamic structure factor (in the Gaussian approximation), etc. The interplay of the three characteristic times in the problem (the mean residence and flight times, and the period of the ‘local mode’) is elucidated. The emergence of a number of striking features of oscillatory diffusion (e.g., the local mode peak in the dynamic mobility and structure factor, and the transition between the oscillatory and diffusive regimes) is demonstrated.     

Random walk in distorted crystal structure

The problem of random walk of particles over the lattice points of a crystal is considered. The concepts of characteristic times of jumps are used in writing and finding the exact solution for a time-dependent equation of migration in a one-dimensional structure with an anisotropic probability of jumps and in the presence of concentrated traps. The case of some characteristic values of the particle capture efficiency of the trap is considered. Differences are revealed between the results of microscopic analysis and the corresponding results of macroscopic theories and the character of the approximation to the deductions of these theories are shown. Some results are also found for steady-state migration and for a three-dimensional structure.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 121–125, February, 1978.     

Random walk statistics on fractal structures

We consider some statistical properties of simple random walks on fractal structures viewed as networks of sites and bonds: range, renewal theory, mean first passage time, etc. Asymptotic behaviors are shown to be controlled by the fractal (¯d) and spectral (¯d) dimensionalities of the considered structure. A simple decimation procedure giving the value of (¯d) is outlined and illustrated in the case of the Sierpinski gaskets. Recent results for the trapping problem, the self-avoiding walk, and the true-self-avoiding walk are briefly reviewed. New numerical results for diffusion on percolation clusters are also presented.     

Solvable models of temporally correlated random walk on a lattice

We seek the conditional probability functionP(m,t) for the position of a particle executing a random walk on a lattice, governed by the distributionW(n, t) specifying the probability ofn jumps or steps occurring in timet. Uncorrelated diffusion occurs whenW is a Poisson distribution. The solutions corresponding to two different families of distributionsW are found and discussed. The Poissonian is a limiting case in each of these families. This permits a quantitative investigation of the effects, on the diffusion process, of varying degrees of temporal correlation in the step sequences. In the first part, the step sequences are regarded as realizations of an ongoing renewal process with a probability densityψ(t) for the time interval between successive jumps.W is constructed in terms ofψ using the continuous-time random walk approach. The theory is then specialized to the case whenψ belongs to the class of special Erlangian density functions. In the second part,W is taken to belong to the family of negative binomial distributions, ranging from the geometric (most correlated) to the Poissonian (uncorrelated). Various aspects such as the continuum limit, the master equation forP, the asymptotic behaviour ofP, etc., are discussed.     

Random walk on a random walk

The authors investigate the random walk of a particle on a one-dimensional chain which has been constructed by a random-walk procedure. Exact expressions are given for the mean-square displacement and the fourth moment after n steps. The probability density after n steps is derived in the saddle-point approximation, for large n. These quantities have also been studied by numerical simulation. The extension to continuous time has been made where the particle jumps according to a Poisson process. The exact solution for the self-correlation function has been obtained in the Fourier and Laplace domain. The resulting frequency-dependent diffusion coefficient and incoherent dynamical structure factor have been discussed. The model of random walk on a random walk is applied to self-diffusion in the concentrated one-dimensional lattice gas where the correct asymptotic behavior is found.     

Chaotic jets with multifractal space-time random walk

The problem of normal and anomalous diffusion is examined for the four-dimensional (4-D) map that arises from the problem of particle motion in a constant magnetic field and electrostatic wave packet. This 4-D map consists of two coupled 2-D maps: a standard map and a web map. The case of a weak chaos is considered. It is shown that due to the finite observation time, the particle diffusion possesses strong nonhomogeneous properties. Existence of long-living bundles of orbits with coherent propagation property is checked. These bundles are named "chaotic jets." The same name is used for a part of the trajectory if this part corresponds to long-living trapping or flight. The existence of chaotic jets depends on the topological properties of the phase space and influences the asymptotic law of transport. The particle transport can be considered as a random walk in the multifractal space-time that is produced by flights and trappings of a test particle in some area of its phase space. Levy random walk theory and its generalization for the multifractal space-time situation is considered and asymptotic laws for displacements are derived. Different intermediate asymptotics are discussed.     

A generalization of random walk models to correlations over two jumps

Using published results on continuous time random walk theories, we show that the random walk theory of Gissler and Rother is equivalent to a master equation with jumps to further neighbor sites. We extend the theory to include time correlations over two jumps. No special assumptions are made in the analysis, so that the theory may be applied to any lattice type with a general time probability distribution for jumps; a generalized second-order differential equation is given for the results. In the special case of an exponential time probability density, a simple homogeneous second order differential equation is obtained which is shown to be equivalent to a certain two-state master equation model.     

The exact probability distribution of a two-dimensional random walk

A calculation is made of the exact probability distribution of the two-dimensional displacement of a particle at timet that starts at the origin, moves in straight-line paths at constant speed, and changes its direction after exponentially distributed time intervals, where the lengths of the straight-line paths and the turn angles are independent, the angles being uniformly distributed. This random walk is the simplest model for the locomotion of microorganisms on surfaces. Its weak convergence to a Wiener process is also shown.     

Analytic study of the effect of persistence on a one-dimensional biased random walk

An analytic study of a one-dimensional biased random walk with correlations between nearest-neighbour steps is presented, both in a lattice model and in its continuous version. First, the treatment of the unbiased problem is recalled and the effect of correlations on the diffusion coefficient is discussed. Then the study is extended to the biased case. The problem is then completely determined by two independent parameters, the degree of correlations in the motion on the one hand and the value of the bias on the other. Both the velocity of the particle and its diffusion coefficient are computed. As a result, the velocity as well as the diffusion coefficient are enhanced when there are positive correlations (qualified as persistence) in the motion, and reduced in the opposite case.     

Random walk on hierarchical comb structures

This paper examines random walks on an exactly solvable comb model of percolation clusters. The study shows that diffusion along the structure’s axis is anomalous. Generalized diffusion equations with fractional-order time derivatives are derived, and a generalization to the multidimensional case is carried out. The relationship between this problem and that of diffusion in a medium with traps is examined, and equations that describe diffusion in a medium with traps are derived. The paper also discusses the transition to ordinary diffusion due to the introduction of comb teeth of finite length, and analyzes the case of N teeth of different length. It is shown that the solution of this problem leads to the emergence of an N-channel diffusion equation. Finally, equations describing the diffusion of interacting electrons are derived. Zh. éksp. Teor. Fiz. 115, 1285–1296 (April 1999)     

The two-state random walk

We develop asymptotic results for the two-state random walk, which can be regarded as a generalization of the continuous-time random walk. The two-state random walk is one in which a particle can be in one of two states for random periods of time, each of the states having different spatial transition probabilities. When the sojourn times in each of the states and the second moments of transition probabilities are finite, the state probabilities have an asymptotic Gaussian form. Several known asymptotic results are reproduced, such as the Gaussian form for the probability density of position in continuous-time random walks, the time spent in one of these states, and the diffusion constant of a two-state diffusing particle.     

Nonstochastic behavior of atomic surface diffusion on Cu(111) down to low temperatures

Atomic diffusion is usually understood as a succession of random, independent displacements of an adatom over the surface's potential energy landscape. Nevertheless, an analysis of molecular dynamics simulations of self-diffusion on Cu(111) demonstrates the existence of different types of correlations in the atomic jumps at all temperatures. Thus, the atomic displacements cannot be correctly described in terms of a random walk model. This fact has a profound impact on the determination and interpretation of diffusion coefficients and activation barriers.     

Two-state random walk model of lattice diffusion. 1. Self-correlation function

Diffusion with interruptions (arising from localized oscillations, or traps, or mixing between jump diffusion and fluid-like diffusion, etc.) is a very general phenomenon. Its manifestations range from superionic conductance to the behaviour of hydrogen in metals. Based on a continuous-time random walk approach, we present a comprehensive two-state random walk model for the diffusion of a particle on a lattice, incorporating arbitrary holding-time distributions for both localized residence at the sites and inter-site flights, and also the correct first-waiting-time distributions. A synthesis is thus achieved of the two extremes of jump diffusion (zero flight time) and fluid-like diffusion (zero residence time). Various earlier models emerge as special cases of our theory. Among the noteworthy results obtained are: closed-form solutions (ind dimensions, and with arbitrary directional bias) for temporally uncorrelated jump diffusion and for the ‘fluid diffusion’ counterpart; a compact, general formula for the mean square displacement; the effects of a continuous spectrum of time scales in the holding-time distributions, etc. The dynamic mobility and the structure factor for ‘oscillatory diffusion’ are taken up in part 2.     

Environment-dependent continuous time random walk

A generalized continuous time random walk model which is dependent on environmental damping is proposed in which the two key parameters of the usual random walk theory: the jumping distance and the waiting time, are replaced by two new ones: the pulse velocity and the flight time. The anomalous diffusion of a free particle which is characterized by the asymptotical mean square displacement <x²(t)>～t^α is realized numerically and analysed theoretically, where the value of the power index α is in a region of 0 < α < 2. Particularly, the damping leads to a sub-diffusion when the impact velocities are drawn from a Gaussian density function and the super-diffusive effect is related to statistical extremes, which are called rare-though-dominant events.     

On the relation between conduction and diffusion in a random walk along self-similar clusters

The relation between diffusion and conduction in the random walk of a particle by means of Lévy hops is investigated. It is shown that on account of the unusual character of Lévy hops, the mobility of a particle is a nonlinear function of the electric field for arbitrarily weak fields. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 7, 518–520 (10 April 1998)     

Application of random walk concept to the cyclic diffusion mechanisms for self-diffusion in intermetallic compounds

Huntington–Elcock–McCombie (HEM) mechanism involving six consecutive and correlated jumps, a triple-defect mechanism (TDM) involving three correlated jumps and an anti-structure bridge (ASB) mechanism invoking the migration of an anti-structure atom are the three mechanisms currently in vogue to explain the self- and solute diffusion in intermetallic compounds. Among them, HEM and TDM are cyclic in nature. The HEM and TDM constitute the theme of the present article. The concept of random walk is applied to them and appropriate expressions for the diffusion coefficient are derived. These equations are then employed to estimate activation energies for self-diffusion via HEM and TDM processes and compared with the available experimental data on activation energy for self-diffusion in intermetallic compounds. The resulting activation energies do not favour HEM and TDM for the self-diffusion in intermetallic compounds. A comparison of the sum of experimentally determined activation energies for vacancy formation and migration with the activation energies for self-diffusion determined from radioactive tracer method favours the conventional monovacancy-mediated process for self-diffusion in intermetallic compounds.     

Mechanisms and energetic of atomic processes in surface diffusion

Surface diffusion is a subject of basic importance for understanding mass transport phenomena in surface and nano science. In the particle aspect of surface diffusion of single atoms and simple molecules, information of interest is the detail atomic mechanisms and the activation energy of various atomic processes, and also the binding energy of atoms at different surface sites. In the absence of an external force, atoms will perform random walk without a preferred direction. When an atom is subjected to an external force, or when a chemical potential gradient exists, it will move preferentially in the direction of the force, or in the direction of decreasing chemical potential, thus the random walk becomes directional. Using atomic resolution microscopy, it is now possible to observe random walk diffusion of atoms, molecules and atomic clusters directly as well as to study the dynamic behavior of atoms as perturbed by the electronic interactions of the surface in great detail. Here, methods of studying quantitatively the particle aspect of surface diffusion and how it affects the dynamic behavior of the surface are very briefly reviewed.     

Random walk model with correlated jumps: Self-correlation function and frequency-dependent diffusion coefficient

In this paper we report the results of a generalization of the continuous-time random walk theory of Montroll and Weiss to include correlations over two jumps. We expand our results of a previous letter by discussing the multi-state master equation formulation for Bravais and non-Bravais lattices. We present results for these equations and discuss the ease of tetrahedral interstitial sites in body centered cubic lattices which have a realization in hydrogen in metal systems. We prove that the Bravais lattices always reduce to a second-order differential equation and we give a general solution for the frequency-dependent diffusion coefficient in this case.     

Diffusion and relaxation of the fractional order in fractal media in the classical and quantum cases

Two model examples of the application of fractional calculus are considered. The Riemann–Liouville fractional derivative with 0 < α ≤ 1 was used. The solution of a fractional equation, which describes anomalous relaxation and diffusion in an isotropic fractal space, has been obtained in the form of the product of a Fox function by a Mittag-Leffler function. The solution is simpler than that given in Ref. 6 and it generalizes the result reported in Ref. 7. For the quantum case, a solution of the generalized Neumann–Kolmogorov fractional quantum-statistical equation has been obtained for an incomplete statistical operator which describes the random walk of a quantum spin particle, retarded in traps over a fractal space. The solution contains contributions from quantum Mittag-Leffler (nonharmonic) fractional oscillations, anomalous relaxation, noise fractional oscillations, and exponential fractional diffusion oscillation damping.