Author

# P. F. J. van Velthoven

Bio: P. F. J. van Velthoven is an academic researcher from Utrecht University. The author has contributed to research in topics: Random walk & Square lattice. The author has an hindex of 1, co-authored 1 publications receiving 18 citations.

Topics: Random walk, Square lattice, Sigma

##### Papers

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TL;DR: In this article, the authors apply the methods of kinetic theory to obtain systematic expansions of dc and ac transport properties in powers of the impurity concentration c. The method is applied to a hopping model on a d-dimensional cubic lattice having two types of bonds with conductivity sigma and sigma/sub 0/ = 1, with concentrations c and 1-c, respectively.

Abstract: The authors consider diffusive systems with static disorder, such as Lorentz gases, lattice percolation, ants in a labyrinth, termite problems, random resistor networks, etc. In the case of diluted randomness the authors can apply the methods of kinetic theory to obtain systematic expansions of dc and ac transport properties in powers of the impurity concentration c. The method is applied to a hopping model on a d-dimensional cubic lattice having two types of bonds with conductivity sigma and sigma/sub 0/ = 1, with concentrations c and 1-c, respectively. For the square lattice the authors explicitly calculate the diffusion coefficient D(c,sigma) as a function of c, to O(c/sup 2/) terms included for different ratios of the bond conductivity sigma. The probability of return at long times is given by P/sub 0/(t) approx. (4..pi..D(c,sigma)t)/sup -d/2/, which is determined by the diffusion coefficient of the disordered system.

18 citations

##### Cited by

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TL;DR: In this article, the authors present a review of continuous-time random walk theory for diffusion of single particles on lattices with frozen-in disorder, including models with regular transition rates and irregular lattices.

835 citations

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TL;DR: In this paper, the authors review theoretical and experimental studies of the AC dielectric response of inhomogeneous materials, modelled as bond percolation networks, with a binary (conductor-dielectric) distribution of bond conductances.

Abstract: We review theoretical and experimental studies of the AC dielectric response of inhomogeneous materials, modelled as bond percolation networks, with a binary (conductor-dielectric) distribution of bond conductances. We first summarize the key results of percolation theory, concerning mostly geometrical and static (DC) transport properties, with emphasis on the scaling properties of the critical region around the percolation threshold. The frequency-dependent (AC) response of a general binary model is then studied by means of various approaches, including the effective-medium approximation, a scaling theory of the critical region, numerical computations using the transfer-matrix algorithm, and several exactly solvable deterministic fractal models. Transient regimes, related to singularities in the complex-frequency plane, are also investigated. Theoretical predictions are made more explicit in two specific cases, namely R-C and RL-C networks, and compared with a broad variety of experimental resul...

562 citations

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TL;DR: The current paper demonstrates that this procedure can be extended to elliptic difference equations defined on infinite lattices, or on finite lattice with boundary conditions of either Dirichlet or Neumann type.

18 citations

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TL;DR: For random walks on two and three-dimensional cubic lattices, numerical results are obtained for the static,D(∞), and time-dependent diffusion coefficientD(t), as well as for the velocity autocorrelation function (VACF) as discussed by the authors.

Abstract: For random walks on two- and three-dimensional cubic lattices, numerical results are obtained for the static,D(∞), and time-dependent diffusion coefficientD(t), as well as for the velocity autocorrelation function (VACF). The results cover all times and include linear and quadratic terms in the density expansions. Within the context of kinetic theory this is the only model in two and three dimensions for which the time-dependent transport properties have been calculated explicitly, including the long-time tails.

15 citations