By Sidney Redner

First-passage houses underlie a variety of stochastic methods, comparable to diffusion-limited development, neuron firing, and the triggering of inventory techniques. This booklet presents a unified presentation of first-passage tactics, which highlights its interrelations with electrostatics and the ensuing strong effects. the writer starts off with a contemporary presentation of primary conception together with the relationship among the career and first-passage chances of a random stroll, and the relationship to electrostatics and present flows in resistor networks. the results of this conception are then built for easy, illustrative geometries together with the finite and semi-infinite durations, fractal networks, round geometries and the wedge. numerous purposes are offered together with neuron dynamics, self-organized criticality, diffusion-limited aggregation, the dynamics of spin platforms, and the kinetics of diffusion-controlled reactions. Examples mentioned contain neuron dynamics, self-organized criticality, kinetics of spin platforms, and stochastic resonance.

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Laplacian Formalism Another useful version of the correspondence between diffusion and electrostatics is based on encoding the initial condition as the spatial argument of an electrostatic potential rather than as an initial condition. This Laplacian approach gives the splitting probability for composite domain boundaries in a natural fashion and also provides conditional hitting times, namely, the times to eventually hit a specific subset of the boundary. 1. Splitting Probabilities For simplicity, we4tart with a symmetric nearest-neighbor random walk in the finite interval [x_ x + 1 and then take the continuum limit after developing the formalism.

Biased Diffusion We now consider the role of bias on first-passage characteristics. As a rule, the bias dominates first-passage properties for sufficiently long systems or for long times. This crossover from isotropic to biased behavior is naturally quantified by the Peclet number. Although qualitative features of this crossover can be obtained by intuitive physical arguments, the complete solution reveals many subtle and interesting properties. Absorption Mode.

X_). 4x+ ). ). 0, £ + (x+ ) =, 1. In the continuum limit, Eq. 15) reduces to the one-dimensional Laplace equation EUx) 0. ,(L) 1 on the exit subset of the absorbing boundary and e_ = 0 on the complement of this boundary. These conditions are interchanged fore + . The functions are harmonic because e(x) equals the average of at neighboring points [Eq. 14)1; that is, e, is in "harmony" with its local environment. This is a basic feature of solutions to the Laplace equation. Because e ± satisfies the Laplace equation, we can transcribe well-known results from electrostatics to less familiar, but corresponding, first-passage properties.