By Sidney Redner
First-passage homes underlie quite a lot of stochastic methods, reminiscent of diffusion-limited progress, neuron firing, and the triggering of inventory ideas. This booklet offers a unified presentation of first-passage techniques, which highlights its interrelations with electrostatics and the ensuing robust effects. the writer starts with a contemporary presentation of primary thought 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. a variety of 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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Extra resources for A Guide to First-Passage Processes
1. 6. Connection between First-Passage and Electrostatics 31 We again treat a symmetric random walk on the finite interval [x_, the generalization to higher dimensions and to general hopping processes is straightforward. Let the time increment between successive steps be St, and let t(x) denote the mean time to exit at either boundary component when a particle starts at x. This quantity is simply the time for each exit path times the probability of the path, averaged over all particle trajectories.
As we shall see in the next chapters, this device greatly simplifies the computation of the mean exit times and also provides useful physical insights. First-Passage Fundamentals 32 Finally, we may extend this electrostatic formalism to the conditional exit times in which we discriminate the exit time by which part of the boundary is reached. Thus let t_ (x) be the conditional mean exit time for a random walk that starts at x and exits at B. without hitting B. Similarly let t+ (x) be the conditional mean exit time for a random walk that starts at x and exits at B + without hitting B_.
We divide the boundary points into two classes, B+ and B. _ as functions of the initial position x of the random walk. 4.. , and vice versa for E_. This Laplace equation has a simple resistor network interpretation. The basic connection is that if all the lattice bonds are viewed as resistors (not necessarily identical), then Kirchhoff's laws for steady current flow in the 01) Fig. 5. (a) A lattice graph with boundary sites ,13 + or B_. (13) Corresponding resistor network in which each bond (with rectangle) is a t-E2 resistor.
A Guide to First-Passage Processes by Sidney Redner