By Sidney Redner
First-passage houses underlie quite a lot of stochastic tactics, comparable to diffusion-limited progress, neuron firing, and the triggering of inventory techniques. This e-book presents a unified presentation of first-passage methods, which highlights its interrelations with electrostatics and the ensuing strong effects. the writer starts off with a latest presentation of primary conception together with the relationship among the career and first-passage possibilities of a random stroll, and the relationship to electrostatics and present flows in resistor networks. the implications 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. 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 contain neuron dynamics, self-organized criticality, kinetics of spin platforms, and stochastic resonance.
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Extra resources for A Guide to First-Passage Processes
Once the diffusive spread of the concentration becomes comparable to the interval size, the flux of probability through the boundaries becomes significant and a rapid decay of the density inside the interval ensues. We determine this long-time decay for both isotropic and biased diffusion. 1. 2) This equation is defined on 0 < x < L. subject to the absorbing boundary conditions c(0, t) c(L 0. For simplicity, we consider the initial condition c(x. t 0) — x0) with 0 < xo < L, corresponding to a particle starting at x0.
2. Time-Dependent Formulation 51 Laplace transform is e -t ts + dt where the lower limit accounts crudely for the short-time cutoff in j (t). This cutoff allows us to use the long-time form for j (t) in the integral with negligible errors in determining the small-s behavior of f(s). To compute the asymptotic behavior of j(s) in this example, we rewrite the above integral in the dimensionless form, ) = (s /r )" u'e' du, which has two limiting behaviors as s 0. If s >> I Jr, then j(s) I /r, then j(s) varies linearly with s.
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. Some of these simplifications are also discussed in Gardiner (1985) and Risken (1988). 2. The Green's Function Formalism Let us recall the conventional method to compute the first-passage probability of diffusion.
A Guide to First-Passage Processes by Sidney Redner