Random walks and Schrödinger's equation in dimensions

Transparent Boundary Condition for the Parabolic Equation Modeled by the 4RW

“…A direct application of stochastic techniques to the PE as done in [3] necessitates analytical continuation of fields and boundary data, only possible for very limited problems, that is highly undesirable. The four-state random walk (4RW) model, originally developed by Ord [12] to provide a macroscopic model for the Schrödinger equation, alleviates this limitation and results in the same second-order accurate discretization scheme for the PE as the Crank-Nicolson scheme. Numerical schemes are currently being developed using the 4RW model to study wireless propagation problems in open domains, and it is the purpose of this letter to derive transparent boundary conditions for this model by considering 2D propagation in a 1 1-dimensional space.…”

“…For simplicity and no loss in generality, we first consider the case of and discuss the solution for the nonzero case at the end. For treatment by stochastic methods, (1) is discretized as per the 4RW model, which, like the Crank-Nicolson finite-difference scheme, results in a second-order accurate (in space and time) scheme [12]. For a particle moving on a discrete lattice and subject to random collisions, the transitional probabilities in the 4RW at the discrete space-time point are of the form [7], [12] (2)…”

“…The combination of two directions of motion and two states of spin constitutes the four states in the model. It has been shown in [12] that such a 4RW encompasses the diffusion as well as Schrödinger equations and that is the Schrödinger wave function in the discrete case. The relationship between the temporal and spatial steps of the form implied in the 4RW model assures the aforementioned second-order accuracy in space.…”

Classical spin in a potential field

“…There are now several such systems available, and in these models the objects of study are ensembles of classical particles where the systems exhibit quantum dynamics as secondorder effects. The nonrelativistic free particle in 1 + 1 dimensions is considered in (Ord, 1996a) and Ord and Deakin (1996), and a 16-state (2 + 1)-dimensional model may be found in Ord and Deakin (1997). Relativistic versions may be found in Ord (1996b, c).…”

“…The model uses only a four-state history and is thus simpler than a previous 16-state version (Ord and Deakin, 1997).…”