By Sidney Redner
First-passage homes underlie a variety of stochastic techniques, similar to diffusion-limited progress, neuron firing, and the triggering of inventory ideas. This e-book presents a unified presentation of first-passage procedures, which highlights its interrelations with electrostatics and the ensuing robust results. the writer starts off with a contemporary 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 concept 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 functions are offered together with neuron dynamics, self-organized criticality, diffusion-limited aggregation, the dynamics of spin platforms, and the kinetics of diffusion-controlled reactions. Examples mentioned contain neuron dynamics, self-organized criticality, kinetics of spin structures, and stochastic resonance.
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Additional resources for A Guide to First-Passage Processes
Equation for the mean exit, or mean first-passage, time. These results can be extended to a general short-range hopping process with single-step hopping probability N,Fi. In this case, the analog of Eq. 622) which in the continuum limit, becomes the Poisson-like equation DV 2 t(1 - ) + ii(F) • 't(F) = —1. 23) Note, in particular, that the determination of the mean exit time has been recast as a time-independent electrostatic problem. 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.
X) by summing the probabilities for all paths that start at x and reach x_ without touching x. (Fig. 4). A parallel statement holds for e + (x). 13) P± where P (x) denotes the probability of a path from x to x + that avoids X T . As illustrated in Fig. 6. Connection between First-Passage and Electrostatics B+ B_ B + B = 2 X 2 B Ai\ANAC`" XX ' Fig. 1,4. Decomposition of random-walk paths from x to the absorbing subset B (the point x_ ) into the outcome after one step and the remainder from x' to 8- The factors 1 /2 account for the probabilities associated with the first step of the decomposed paths.
This behavior is more commonly known as transient [Nilya (1921) and Feller (1968)]. The distinction between recurrence and transience has important physical implications. As a basic example, consider a diffusion-controlled reaction in three dimensions. Because the trajectory of each diffusing reactant is transparent, the probability for two molecules to meet is essentially independent of their initial separation. This means that a molecule reacts with any other reactant molecule in the system with almost equal probability.