By Sidney Redner
First-passage houses underlie quite a lot of stochastic methods, corresponding to diffusion-limited development, neuron firing, and the triggering of inventory concepts. This e-book offers a unified presentation of first-passage strategies, which highlights its interrelations with electrostatics and the ensuing robust results. the writer starts with a contemporary presentation of primary concept 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 idea are then built for easy, illustrative geometries together with the finite and semi-infinite durations, fractal networks, round geometries and the wedge. quite a few functions 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 structures, and stochastic resonance.
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Additional info for A Guide to First-Passage Processes
X _ ) ( } 0_ [p p_(x- - 6x)(t p _(x _ 6x) ± 6t) + Pp_ (x Sx)(t p _(x Sx) Bt)] 1 = St _(x) + — te _(x — Sx)t_(x — Sx) 2 £_(x Sx)t_(x +Sx)] St £_(x) +16 _(x)t_(x)+ . 2C 2 8 2 5_, (X ):_(X)] s: ax . 26) with D , 2(Br-) 2 /26/ and subject to the boundary conditions e _(r7)t_o 0 both on (where t_ vanishes) and on B + (where &_ vanishes). The governing equations and boundary conditions for th () are entirely analogous. Finally, if there is a bias in the hopping process, then Eq. :)4(1:)] - e±o. 27) With this formalism, we can obtain eventual hitting probabilities and mean hitting times (both unconditional and conditional) by solving timeindependent electrostatic boundary-value problems.
This leads to the analog of Eq. 19) where t p (x) is the exit time of a specific path to the boundary that starts at x. In analogy with Eq. 20) with the boundary conditions t(x_ ) t(x + ) = 0, which correspond to the exit time being equal to zero if the particle starts at the boundary. This recursion relation expresses the mean exit time starting at x in terms of the outcome one step in the future, for which the initial walk can be viewed as restarting at either x + 8x or at x — 8x, each with probability 1/2, but also with the time incremented by (St.
15) where A (2) is the discrete second-difference operator, which is defined by ❑ (2) f (x) f (x — 6x) — 2 f (x) f (x 6x). Note the opposite sense ofthis recursion formula compared with master equation Eq. 1) for the probability distribution. Here e f (x) is expressed in terms of output from x, whereas in the master equation, the occupation probability at x is expressed in terms of input to x. ,(x_). 4x+ ). ). 0, £ + (x+ ) =, 1. In the continuum limit, Eq. 15) reduces to the one-dimensional Laplace equation EUx) 0.