A Combinatorial Approach to Discrete Diffusion on a Segment

Canadian Journal of Physics, 2004

The one-dimensional random walk between two reflecting walls is considered from two different points of view: the first, as a particular case of jumps between neighbouring, discrete states; the second, as a system that obeys a generalized diffusion equation. By performing the suitable limits, the identity of the two results is pointed out and a physical application is presented. PACS Nos.: 02.50.Ey, 02.30.Jr

Journal of Computational Physics, 2007

A general methodology for the solution of partial differential equations is described in which the discretization of the calculus is exact and all approximation occurs as an interpolation problem on the material constitutive equations. The fact that the calculus is exact gives these methods the ability to capture the physics of PDE systems well. The construction of both node and cell based methods of first and second order are described for the problem of unsteady heat conduction-though the method is applicable to any PDE system. The performance of these new methods are compared to classic solution methods on unstructured 2D and 3D meshes for a variety of simple and complex test cases.

SIAM Journal on Applied Mathematics, 2008

The radiation (reactive or Robin) boundary condition for the diffusion equation is widely used in chemical and biological applications to express reactive boundaries. The underlying trajectories of the diffusing particles are believed to be partially absorbed and partially reflected at the reactive boundary; however, the relation between the reaction constant in the Robin boundary condition and the reflection probability is not well defined. In this paper we define the partially reflected process as a limit of the Markovian jump process generated by the Euler scheme for the underlying Itô dynamics with partial boundary reflection. Trajectories that cross the boundary are terminated with probability P √ Δt and otherwise are reflected in a normal or oblique direction. We use boundary layer analysis of the corresponding master equation to resolve the nonuniform convergence of the probability density function of the numerical scheme to the solution of the Fokker-Planck equation in a half-space, with the Robin constant κ. The boundary layer equation is of the Wiener-Hopf type. We show that the Robin boundary condition is recovered if and only if trajectories are reflected in the conormal direction σn, where σ is the (possibly anisotropic) constant diffusion matrix and n is the unit normal to the boundary. Otherwise, the density satisfies an oblique derivative boundary condition. The constant κ is related to P by κ = rP √ σn, where r = 1/ √ π and σn = n T σn. The reflection law and the relation are new for diffusion in higher dimensions.

The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 1982

The discrete random walk problem for the unrestricted particle formulated in the double diffusion model given in Hill [2] is solved explicitly. In this model it is assumed that a particle moves along two distinct horizontal paths, say the upper path I and lower path 2. For i = 1, 2, when the particle is in path i, it can move at each jump in one of four possible ways, one step to the right with probability pi, one step to the left with probability qi, remains in the same position with probability ri, or exchanges paths but remains in the same horizontal position with probability si (pi + qi + ri + si = 1). Using generating functions, the probability distribution of the position of an unrestricted particle is derived. Finally some special cases are discussed to illustrate the general result.

Computers & Mathematics with Applications, 2003

SSRN Electronic Journal, 2014

This paper investigates the position (state) distribution of the single step binomial (multi-nomial) process on a discrete state / time grid under the assumption that the velocity process rather than the state process is Markovian. In this model the particle follows a simple multi-step process in velocity space which also preserves the proper state equation of motion. Many numerical numerical examples of this process are provided. For a smaller grid the probability construction converges into a correlated set of probabilities of hyperbolic functions for each velocity at each state point. It is shown that the two dimensional process can be transformed into a Telegraph equation and via transformation into a Klein-Gordon equation if the transition rates are constant. In the last Section there is an example of multi-dimensional hyperbolic partial differential equation whose numerical average satisfies Newton's equation. There is also a momentum measure provided both for the two-dimensional case as for the multi-dimensional rate matrix.

Given a doubly infinite sequence of positive numbers {c k : k ∈ Z} such that {c −1 k : k ∈ Z} satisfies a LLN with limit α ∈ (0, ∞], we consider the nearest-neighbor simple exclusion process on Z where c k is the probability rate of jumps between k and k + 1. If α = ∞ we require an additional minor technical condition. By extending a method developed in [11] we show that the diffusively rescaled process has hydrodynamic behavior described by the heat equation with diffusion constant 1/α. In particular, the process has diffusive behavior for α < ∞ and subdiffusive behavior for α = ∞.

Journal of Physics A: Mathematical and General, 1998

We study, on a d dimensional hypercubic lattice, a random walk which is homogeneous except for one site. Instead of visiting this site, the walker hops over it with arbitrary rates. The probability distribution of this walk and the statistics associated with the hop-overs are found exactly. This analysis provides a simple approach to the problem of tagged diffusion, i.e., the movements of a tracer particle due to the diffusion of a vacancy. Applications to vacancy mediated disordering are given through two examples.

Central European Journal of Mathematics, 2006

We have random number of independent diffusion processes with absorption on boundaries in some region at initial time t = 0. The initial numbers and positions of processes in region is defined by Poisson random measure. It is required to estimate of number of the unabsorbed processes for the fixed time τ > 0. The Poisson random measure depends on τ and τ → ∞.