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Diffusion in Regular and Disordered Lattices

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Abstract

Classical diffusion of single particles on lattices with frozen-in disorder is surveyed. The methods of continuous-time random walk theory are pedagogically developed and applications to solid-state physics are discussed. The first part of the review treats models with regular transition rates; these models possess internal structure or correlations over two jumps and complete solutions are given for each case. The second part of the review covers models with disorder in the transition rates and irregular lattices. For these problems too, explicit calculations and methods are explained and discussed.
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... Two limiting cases will be considered in detail: Random barriers which modify only the adatom mobility, and random traps for which binding energies and diffusion barriers are modified by the same amount (see Fig.1). Random barriers and random traps are standard models in the theory of diffusion in disordered media [10]. ...
... This corresponds to the random barrier energy landscape illustrated in Fig.1 (a), where the binding energies remain unaffected while the diffusion barriers are modified by an amount ∆E D , which can be positive (as in Fig.1 (a)) or negative. An exact analytic expression for the effective diffusion coefficient in two dimensions is not available for the random barrier model [10], but some general conclusions can be drawn from Eqs. (8,9), which are to be evaluated with µ eff = 0. Since D(x) is a monotonic function of x, we have that l/D(0) ≤ M 0 ≤ l/D(l), l/2D(0) ≤ M 1 ≤ l/2D(l) and M 1 /M 0 ≥ 1/2 for ∆E D > 0, and the converse inequalities for ∆E D < 0. Using these relations it is straightforward to prove that attachment is primarily to the descending step (j − ≥ F l/2 ≥ j + ) when ∆E D > 0, and to the ascending step when ∆E D < 0, for both types of boundary conditions. ...
... which is exact for the one-dimensional random barrier model [10]. Here D 0 is the diffusion coefficient on the clean surface, and b = φ(e β∆ED −1)/l is a dimensionless parameter describing the strength and the sign of the mobility gradient( 2 ). ...
Preprint
Codeposition of impurities during the growth of a vicinal surface leads to an impurity concentration gradient on the terraces, which induces corresponding gradients in the mobility and the chemical potential of the adatoms. Here it is shown that the two types of gradients have opposing effects on the stability of the surface: Step bunching can be caused by impurities which either lower the adatom mobility, or increase the adatom chemical potential. In particular, impurities acting as random barriers (without affecting the adatom binding) cause step bunching, while for impurities acting as random traps the combination of the two effects reduces to a modification of the attachment boundary conditions at the steps. In this case attachment to descending steps, and thus step bunching, is favored if the impurities bind adatoms more weakly than the substrate.
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Preprint
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... There is a large body of work on random searchers diffusing in a sea of stationary, randomly distributed targets of density n and binding to any one target where they get trapped [9][10][11][12][13][14][15][16][17]. For large times, the survival probability P s (t ) of a searcher, i.e., the probability that the searcher has not been trapped after time t, has been shown to follow a stretched exponential exp(−k λ 2/(d+2) t d/(d+2) ) [17,18]. ...
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Full-text available
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... It is worth stressing that the apparent simplicity of Eq. (1.7) is deceptive. In fact, the relation between the search process on the phase space Ω, encoded via the matrix Π, and its spectral properties, is sophisticated [123,124]. Further details and examples of discrete-space search processes are presented in Chapter 5 for multi-target search in bounded and heterogeneous environments, and in Chapter 21 for conservative and non-conservative diffusion towards a target in a networked environment. ...
Chapter
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... The theory of renormalization groups has found numerous applications in various fields of physics, such as in the physics of magnetism [17,18], the description of the conductivity of disordered channels [19][20][21][22], coherent ideal absorbers [23], and the mechanical properties of chaotic soft matter [24,25]. On the one hand, each of the above problems has some important symmetries of its own, which lead to different classes of statistical ensembles. ...
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