By Sidney Redner

First-passage houses underlie a variety of stochastic tactics, akin to diffusion-limited progress, neuron firing, and the triggering of inventory recommendations. This e-book presents a unified presentation of first-passage methods, which highlights its interrelations with electrostatics and the ensuing robust outcomes. the writer starts with a latest presentation of primary conception together with the relationship among the career and first-passage chances 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 periods, fractal networks, round geometries and the wedge. numerous 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 contain neuron dynamics, self-organized criticality, kinetics of spin structures, and stochastic resonance.

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**Sample text**

Depending on the nature of the boundary conditions, there are three generic cases, which we term (i) abAorption mode, (ii) transmission mode, and (iii) reflection mode. (i) Absorption Mode: Both boundaries are absorbing. The following are the basic first-passage questions: • What is the time dependence of the survival probability S(t)? This is the probability that a diffusing particle does not touch either absorbing boundary before time t. • What is the time dependence of the first-passage probabilities, or the exit probabilities, to either 0 or to L as a function of the starting position of the particle?

Because the functions sin[(nnx)/Lj are a complete set of states for the given boundary conditions, any initial condition can be represented as an expansion of this form. Each eigenmode decays exponentially in time, with a numerically different L 2 it n 22 D. Because the large-n eigenmodes decay decay time rn more rapidly with time, only the most slowly decaying eigenmode remains in the long-time limit. 4) As we shall see, the longest decay time dynamics within the absorbing interval. 2. L2. 5) at ax2 The presence of the convection term fundamentally alters the asymptotic behavior of diffusion, with the competition between convection and diffusion leading to a subtle crossover in the behavior of the survival probability.

We can simplify the algebra considerably by choosing the appropriate linear combination of exponential solutions that manifestly satisfies the boundary conditions at the outset. The absorbing boundary condition at x = 0 suggests an antisymmetric combination of exponentials. Further, the form of the Green's function as x L must be identical to that as x 0. 7) for the subdomain Green's functions c, and c,, where A and B are constants. 8) with x, max(x x 0 ) and x, min(x, xo). This notation allows us to write the Green's function in the entire domain as a single expression.