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Register a new userVacancy diffusion in the Cu(001) surface II: Random walk theory
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Authors
E. Somfai
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We develop a version of the vacancy mediated tracer diffusion model, which follows the properties of the physical system of In atoms diffusing within the top layer of Cu(001) terraces. This model differs from the classical tracer diffusion problem in that (i) the lattice is finite, (ii) the boundary is a trap for the vacancy, and (iii) the diffusion rate of the vacancy is different, in our case strongly enhanced, in the neighborhood of the tracer atom. A simple continuum solution is formulated for this problem, which together with the numerical solution of the discrete model compares well with our experimental results.
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We have used the indium/copper surface alloy to study the dynamics of surface vacancies on the Cu(001) surface. Individual indium atoms that are embedded within the first layer of the crystal, are used as probes to detect the rapid diffusion of surface vacancies. STM measurements show that these indium atoms make multi-lattice-spacing jumps separated by long time intervals. Temperature dependent waiting time distributions show that the creation and diffusion of thermal vacancies form an Arrhenius type process with individual long jumps being caused by one vacancy only. The length of the long jumps is shown to depend on the specific location of the indium atom and is directly related to the lifetime of vacancies at these sites on the surface. This observation is used to expose the role of step edges as emitting and absorbing boundaries for vacancies.
A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. The fundamental solution (for the {Cauchy} problem) of the fractional diffusion equations can be interpreted as a probability density evolving in time of a peculiar self-similar stochastic process that we view as a generalized diffusion process. By adopting appropriate finite-difference schemes of solution, we generate models of random walk discrete in space and time suitable for simulating random variables whose spatial probability density evolves in time according to a given fractional diffusion equation.
Expanding media are typical in many different fields, e.g. in Biology and Cosmology. In general, a medium expansion (contraction) brings about dramatic changes in the behavior of diffusive transport properties. Here, we focus on such effects when the diffusion process is described by the Continuous Time Random Walk (CTRW) model. For the case where the jump length and the waiting time probability density functions (pdfs) are long-tailed, we derive a general bifractional diffusion equation which reduces to a normal diffusion equation in the appropriate limit. We then study some particular cases of interest, including Levy flights and subdiffusive CTRWs. In the former case, we find an analytical exact solution for the Greens function (propagator). When the expansion is sufficiently fast, the contribution of the diffusive transport becomes irrelevant at long times and the propagator tends to a stationary profile in the comoving reference frame. In contrast, for a contracting medium a competition between the spreading effect of diffusion and the concentrating effect of contraction arises. For a subdiffusive CTRW in an exponentially contracting medium, the latter effect prevails for sufficiently long times, and all the particles are eventually localized at a single point in physical space. This Big Crunch effect stems from inefficient particle spreading due to subdiffusion. We also derive a hierarchy of differential equations for the moments of the transport process described by the subdiffusive CTRW model. In the case of an exponential expansion, exact recurrence relations for the Laplace-transformed moments are obtained. Our results confirm the intuitive expectation that the medium expansion hinders the mixing of diffusive particles occupying separate regions. In the case of Levy flights, we quantify this effect by means of the so-called Levy horizon.
Levy walk process is one of the most effective models to describe superdiffusion, which underlies some important movement patterns and has been widely observed in the micro and macro dynamics. From the perspective of random walk theory, here we investigate the dynamics of Levy walks under the influences of the constant force field and the one combined with harmonic potential. Utilizing Hermite polynomial approximation to deal with the spatiotemporally coupled analysis challenges, some striking features are detected, including non Gaussian stationary distribution, faster diffusion, and still strongly anomalous diffusion, etc.
In a simple model of a continuous random walk a particle moves in one dimension with the velocity fluctuating between V and -V. If V is associated with the thermal velocity of a Brownian particle and allowed to be position dependent, the model accounts readily for the particles drift along the temperature gradient and recovers basic results of the conventional thermophoresis theory.
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