By Sidney Redner

ISBN-10: 0521036917

ISBN-13: 9780521036917

ISBN-10: 0521652480

ISBN-13: 9780521652483

First-passage homes underlie a variety of stochastic methods, comparable to diffusion-limited progress, neuron firing, and the triggering of inventory recommendations. This ebook offers a unified presentation of first-passage approaches, which highlights its interrelations with electrostatics and the ensuing strong outcomes. the writer starts off with a latest presentation of basic idea together with the relationship among the profession and first-passage chances of a random stroll, and the relationship to electrostatics and present flows in resistor networks. the results of this thought are then constructed for easy, illustrative geometries together with the finite and semi-infinite durations, fractal networks, round geometries and the wedge. a variety of purposes are provided together with neuron dynamics, self-organized criticality, diffusion-limited aggregation, the dynamics of spin structures, and the kinetics of diffusion-controlled reactions. Examples mentioned comprise neuron dynamics, self-organized criticality, kinetics of spin structures, and stochastic resonance.

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**Example text**

Because the voltage 1. 7 also equals the probability for the corresponding random walk to reach B + without reaching B -, the term V J P i j is just the probability that a random walk starts at B + , makes a single step to the sites j (with hopping prohabilities P1j ), and then returns to B + without reaching B_. We therefore deduce that ) 1 E g1 E(1 — I 1 J Eg 7 OP+ i 1 +i x (1 — return probability) +1 x escape probability. 4) i Here "escape" means reaching the opposite terminal of the voltage source without returning to the starting point.

Integrating these probabilities, over all time gives the eventual hitting probability to a specified boundary, or equivalently, the splitting probability. What is the dependence of the splitting probability to either 0 or to L as a function of the starting position of the particle? ▪ What is the mean exit time, that is, the mean time until the particle hits either of the absorbing boundaries, again as a function of starting position? What are the conditional exit times, that is, the mean time for the particle to hit a specified houndary as a function of the starting position?

First-Passage Probability and Mean Exit Times At a finer level of description, we investigate not only when a particle is trapped, but also where it is trapped, as embodied by the first-passage probability and moments of the exit time to a specified boundary. 1. Isotropic Diffusion To compute the first-passage probability on the finite interval, it is simpler to use the Laplace transform rather than the Fourier transform because the concentration may be expressed as a single function rather than as an infinite Fourier series.

### A Guide to First-Passage Processes by Sidney Redner

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