By Sidney Redner
First-passage homes underlie quite a lot of stochastic techniques, similar to diffusion-limited progress, neuron firing, and the triggering of inventory concepts. This booklet presents a unified presentation of first-passage strategies, which highlights its interrelations with electrostatics and the ensuing strong results. the writer starts with a latest presentation of primary conception together with the relationship among the profession and first-passage possibilities of a random stroll, and the relationship to electrostatics and present flows in resistor networks. the results of this conception are then constructed for easy, illustrative geometries together with the finite and semi-infinite periods, fractal networks, round geometries and the wedge. quite a few functions are awarded together with neuron dynamics, self-organized criticality, diffusion-limited aggregation, the dynamics of spin platforms, and the kinetics of diffusion-controlled reactions. Examples mentioned comprise neuron dynamics, self-organized criticality, kinetics of spin platforms, and stochastic resonance.
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Additional resources for A Guide to First-Passage Processes
After a sufficiently long time, however, a diffusing particle can reach the reflecting boundary before absorption. The subsequent first-passage properties must also reflect this finite-size cutoff. 40 First Passage in an Interval The first-passage questions for these latter two modes are similar to the those of absorption mode, except that there is only one absorbing boundary: • • • • What is the survival probability S(t)? What is the first-passage probability F(t) to the output? How long does it take the particle to reach the output?
This first step transforms the diffusion equation to the simpler Laplace equation. Then, in computing the flux, the exit probability is just the electric field at the boundary point. 6. Connection between First-Passage and Electrostatics 25 a complete correspondence between a first-passage problem and an electrostatic problem in the same geometry. This mapping is simple yet powerful, and can be adapted to compute related time-integrated properties, such as the splitting probabilities and the moments of the exit time.
This gives ? e ± (x) p± [e+(x A Xl x 29 p±(x (Sx) + — T r± pc 2 — ax) + 6x) + £l (x — 6x)1. 15) where A (2) is the discrete second-difference operator, which is defined by ❑ (2) f (x) f (x — 6x) — 2 f (x) f (x 6x). Note the opposite sense ofthis recursion formula compared with master equation Eq. 1) for the probability distribution. Here e f (x) is expressed in terms of output from x, whereas in the master equation, the occupation probability at x is expressed in terms of input to x. ,(x_). 4x+ ).