By Sidney Redner
First-passage homes underlie quite a lot of stochastic strategies, corresponding to diffusion-limited progress, neuron firing, and the triggering of inventory innovations. This ebook offers a unified presentation of first-passage procedures, which highlights its interrelations with electrostatics and the ensuing strong results. the writer starts with a contemporary presentation of primary 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 idea are then built for easy, illustrative geometries together with the finite and semi-infinite periods, fractal networks, round geometries and the wedge. numerous functions are offered 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 platforms, and stochastic resonance.
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Additional resources for A Guide to First-Passage Processes
Suppose that the boundary sites in B+ are fixed at unit potential while the sites in B_ are grounded. The net current at each interior site i of the network must be zero, as there is current input and output only at the boundary. This current conservation condition is E g, (V, j — V 1 ) = 0. 1 ) where g, 1 is the conductance of the bond between sites i and j, V, is the voltage at site i, and the sum pans over the nearest neighbors j of site i. Solving for Vi gives E . 2) where the last step applies for a homogeneous network.
9) is the first-passage probability to a given point, F s = is the probability of eventually hitting this p9int. 5. 4. 1 1} exist, then F (s) in Eq. 10) contains only the Taylor series terms. 12) Thus the Laplace transform is a moment generating function, as it contains all the positive integer moments of the probability distribution F(t). This is one of the reasons why the Laplace transform is such a useful tool for first-passage processes. In summary, the small-s behavior of the Laplace transform, or, equivalently, the z 1 behavior of the generating function, are sufficient to determine the long-time behavior of the function itself.
This crossover between the intermodiate-time power law and the long-time exponential decay can be formulated more generally by an approach similar to that given in Chap. 1 for determining the asymptotics of generating functions. Consider the situation in which the first-passage probability has the generic form J() = t —ce e —Eir with r >> 1, and where j(t) is vanishingly small for t << 1. The power law represents the asymptotic behavior of an infinite system, and the exponential factor represents a finite-size cutoff.