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.     

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)     

Irregular diffusion in the bouncing ball billiard

We call a system bouncing ball billiard if it consists of a particle that is subject to a constant vertical force and bounces inelastically on a one-dimensional vibrating periodically corrugated floor. Here we choose circular scatterers that are very shallow, hence this billiard is a deterministic diffusive version of the well-known bouncing ball problem on a flat vibrating plate. Computer simulations show that the diffusion coefficient of this system is a highly irregular function of the vibration frequency exhibiting pronounced maxima whenever there are resonances between the vibration frequency and the average time of flight of a particle. In addition, there exist irregularities on finer scales that are due to higher-order dynamical correlations pointing towards a fractal structure of this curve. We analyze the diffusive dynamics by classifying the attracting sets and by working out a simple random walk approximation for diffusion, which is systematically refined by using a Green–Kubo formula.     

Subordination pathways to fractional diffusion

The uncoupled Continuous Time Random Walk (CTRW) in one space-dimension and under power law regime is splitted into three distinct random walks: (rw ₁), a random walk along the line of natural time, happening in operational time; (w ₂), a random walk along the line of space, happening in operational time; (rw ₃), the inversion of (rw ₁), namely a random walk along the line of operational time, happening in natural time. Via the general integral equation of CTRW and appropriate rescaling, the transition to the diffusion limit is carried out for each of these three random walks. Combining the limits of (rw ₁) and (rw ₂) we get the method of parametric subordination for generating particle paths, whereas combination of (rw ₂) and (rw ₃) yields the subordination integral for the sojourn probability density in space - time fractional diffusion.     

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.     

Chaotic diffusion and 1/f-noise of particles in two-dimensional solids

We investigate the classical nonlinear dynamics of a particle moving conservatively in a two-dimensional periodic potential. The particle exhibits diffusive motion in the absence of random forces. In a broad range of energies above the potential barrier, the diffusion process is anomalously accelerated and associated with 1/f-noise in the power spectrum of velocity fluctuations. The analysis of Poincaré surfaces of section and the distribution of free paths indicate that the phenomenon is caused by a trapping of orbits in a self-similar hierarchy of nested cantori. We describe a statistical theory for this mechanism in terms of a renewal process and a random walk on a hierarchical lattice.Work supported by Deutsche Forschungsgemeinschaft     

Random Walks and Polymers in the Presence of Quenched Disorder

After a general introduction to the field, we describe some recent results concerning disorder effects on both ‘random walk models’, where the random walk is a dynamical process generated by local transition rules, and on ‘polymer models’, where each random walk trajectory representing the configuration of a polymer chain is associated to a global Boltzmann weight. For random walk models, we explain, on the specific examples of the Sinai model and of the trap model, how disorder induces anomalous diffusion, aging behaviours and Golosov localization, and how these properties can be understood via a strong disorder renormalization approach. For polymer models, we discuss the critical properties of various delocalization transitions involving random polymers. We first summarize some recent progresses in the general theory of random critical points: thermodynamic observables are not self-averaging at criticality whenever disorder is relevant, and this lack of self-averaging is directly related to the probability distribution of pseudo-critical temperatures T _c(i,L) over the ensemble of samples (i) of size L. We describe the results of this analysis for the bidimensional wetting and for the Poland–Scheraga model of DNA denaturation.Conference Proceedings “Mathematics and Physics”, I.H.E.S., France, November 2005     

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.     

Dynamics in dense suspensions of charge-stabilized colloidal particles

The dynamic behavior of charge-stabilized colloidal particles in suspension was studied by photon correlation spectroscopy with coherent X-rays (XPCS). The short-time diffusion coefficient, D(Q) , was measured for volume concentrations φ ⩽ 0.18 and compared to the free particle diffusion constant D₀ and the static structure factor S(Q) . The data show that indirect, hydrodynamic interactions are relevant for the system and hydrodynamic functions were derived. The results are in striking contrast to the predictions of the PA (pairwise-additive approximation) model, but show features typical for a hard-sphere system. The observed mobility is however considerably smaller than the one of a respective hard-sphere system. The hydrodynamic functions can be modelled quantitatively if one allows for an increased effective viscosity relative to the hard-sphere case.     

Anisotropic random walks and asymptotically one-dimensional diffusion onabc-gaskets

Asymptotically one-dimensional diffusion processes are studied on the class of fractals calledabc-gaskets. The class is a set of certain variants of the Sierpiński gasket containing infinitely many fractals without any nondegenerate fixed point of renoramalization maps. While the “standard” method of constructing diffusions on the Sierpiński gasket and on nested fractals relies on the existence of a nondegenerate fixed point and hence it is not applicable to allabc-gaskets, the asymptotically one-dimensional diffusion is constructed on anyabc-gasket by means of an unstable degenerate fixed point. To this end, the generating functions for numbers of steps of anisotropic random walks on theabc-gaskets are analyzed, along the line of the authors' previous studies. In addition, a general stategy of handling random walk sequences with more than one parameter for the construction of asymptotically one-dimensional diffusion is proposed.     

A limit theorem for turbulent diffusion

We show under some specific conditions that the formal diffusion approximation for the motion of a particle in a random velocity field is valid.     

Wigner Surmise for Domain Systems

In random matrix theory the spacing distribution functions p ⁽ⁿ⁾(s) are well fitted by the Wigner surmise and its generalizations. In this approximation the spacing functions are completely described by the behavior of the exact functions in the limits s→0 and s→∞. Most non equilibrium systems do not have analytical solutions for the spacing distribution and correlation functions. Because of that, we explore the possibility to use the Wigner surmise approximation in these systems. We found that this approximation provides a first approach to the statistical behavior of complex systems, in particular we use it to find an analytical approximation to the nearest neighbor distribution of the annihilation random walk.     

Transport Equation Evaluation of Coupled Continuous Time Random Walks

The transport behavior of a migrating particle in a disordered medium is exhibited in the solution of a transport equation derived from a coupled continuous time random walk (CTRW). A core aspect of CTRW is the spectrum of transitions in displacement s and time t, ψ(s,t), that characterizes the disordered system, which determine the transport. In many applications the CTRW approach has successfully accounted for the anomalous or non-Fickian nature of the particle plume propagation based on a power-law dependence ψ(t) in a decoupled p(s)ψ(t) approximation to ψ(s,t). For example, this power-law dependence in t derives from the complex Darcy flow fields in geological formations. Recently, the fully coupled CTRW was analyzed using a particle tracking approach, demonstrating that the decoupled approximation is valid only for a compact distribution of s. In this paper we solve the nonlocal-in-time transport equation with a ψ(s,t) containing a power-law dependence in both s (a Lévy-like distribution) and t, which necessitates the strong s,t coupling. We show enhanced transport behavior (relative to the plume propagation behavior reported in the literature) that derives from the rare large displacements in s (limited by the transition t). The interplay between the two coupled power laws is clearly shown in the changes in the breakthrough curves in the arrival times, dispersion and dependence on the velocity (v=s/t) distribution. Similar enhancements are exhibited in the particle tracking results.     

Magnetoplasticity and diffusion in silicon single crystals

The effect of static magnetic fields on the dynamics of surface dislocation segments, as well as the diffusion mobility of a dopant in silicon single crystals, has been analyzed. It has been experimentally found that the preliminary treatment of p-type silicon plates (the dopant is boron with a concentration of 10¹⁶ cm⁻³) in the static magnetic field (B = 1 T, a treatment time of 30 min) leads to an increase in the mobility of surface dislocation segments. The characteristic times of observed changes (about 80 h) and the threshold dopant concentration (10₁₅ cm⁻³) below which the magneto-optical effect in silicon is not fixed have been determined. It has been found that diffusion processes in dislocation-free silicon are magnetically sensitive: the phosphorus diffusion depth in p-type silicon that is preliminarily aged in the static magnetic field increases (by approximately 20%) compared to the reference samples.     

Vacuum Radiation,Entropy, and Molecular Chaos

Vacuum radiation causes a particle to make a random walk about its dynamical trajectory. In this random walk the root mean square change in spatial coordinate is proportional to t ^1/2, and the fractional changes in momentum and energy are proportional to t ^−1/2, where t is time. Thus the exchange of energy and momentum between a particle and the vacuum tends to zero over time. At the end of a mean free path the fractional change in momentum of a particle in a gas is very small. However, at the end of the mean free path each particle undergoes an interaction that magnifies the preceding change, and the net result is that the momentum distribution of the particles in a gas is randomized in a few collision times. In this way the random action of vacuum radiation and its subsequent magnification by molecular interaction produces entropy increase. This process justifies the assumption of molecular chaos used in the Boltzmann transport equation.     

A Limit Result for a System of Particles in Random Environment

We consider an infinite system of particles in one dimension, each particle performs independent Sinai’s random walk in random environment. Considering an instant t, large enough, we prove a result in probability showing that the particles are trapped in the neighborhood of well defined points of the lattice depending on the random environment, t and the starting points of the particles. Supported by GREFI-MEFI and Departimento di Mathematica, Universita di Roma II “Tor Vergata”, Italy.     

In vivo anomalous diffusion and weak ergodicity breaking of lipid granules

Combining extensive single particle tracking microscopy data of endogenous lipid granules in living fission yeast cells with analytical results we show evidence for anomalous diffusion and weak ergodicity breaking. Namely we demonstrate that at short times the granules perform subdiffusion according to the laws of continuous time random walk theory. The associated violation of ergodicity leads to a characteristic turnover between two scaling regimes of the time averaged mean squared displacement. At longer times the granule motion is consistent with fractional Brownian motion.     

Analysis of price diffusion in financial markets using PUCK model

Based on the new type of random walk process called the potentials of unbalanced complex kinetics (PUCK) model, we theoretically show that the price diffusion in large scales is amplified 2(2+b)^-1 times, where b is the coefficient of quadratic term of the potential. In short time scales the price diffusion depends on the size M of the super moving average. Both numerical simulations and real data analysis of Yen-Dollar rates are consistent with theoretical analysis.     

Crossover from normal diffusion to single-file diffusion of particles in a one-dimensional channel: LJ particles in zeolite zsm-22

The crossover from normal Fickian diffusion, where mean-squared displacement goes as t to single-file diffusion, where <z²> t?^0.5 is studied as function of particle size confined in zeolite zsm-22 using molecular dynamics simulations. The simulation results indicate that the crossover is smooth as the particle size increases and the diffusion reaches single file through a series of subdiffusion processes. A small number of mutual passage of particles can destroy the correlated single-file diffusion. The density dependence on single-file mobility obtained from molecular dynamics simulations are in very good agreement with theoretical predictions.