By Sidney Redner
First-passage homes underlie a variety of stochastic techniques, resembling diffusion-limited development, neuron firing, and the triggering of inventory strategies. This publication 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 basic concept 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 implications of this conception 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 platforms, and the kinetics of diffusion-controlled reactions. Examples mentioned comprise 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
More general initial conditions can easily be considered by the linearity of the basic equations. In terms of C o , the eventual hitting probability 8(F a) is given by AFB, t)dt 8(713). 6) On the other hand, from Eq. 5) this normal derivative is just the electric field associated with the initial charge distribution. This leads to the following fundamental condlusion: • For a diffusiAg particle that is initially at i0 inside a domain with absorbing boundary conditions, the eventual hitting probability to a boundary point F8 equals the electric field at this same location when a point charge of magnitude 1/(D d) is placed at F0 and the domain boundary is grounded.
Note also that the diffusion equation is identical in form to the time-dependent SchrOdinger equation for a quantum particle in the infinite square-well potential, V(x) = 0 for 0 < x < L and V (x) Do otherwise, with the diffusion coefficient D playing the role of h 2 /2m in the Schrodinger equation. 2. Time-Dependent Formulation 41 Therefore the long-time concentration for diffusion in the absorbing interval is closely related to the low-energy eigenstates of the quantum-well system. We solve Eq.
17) f which, in the continuum limit, reduces to if) (F)V 26. 18) where the local diffusion coefficient DV) is just the mean-square displacement and the local velocity 17(F) is the mean displacement alter a single step when starting from P in this hopping process. The existence of the continuum limit requires that the range of the bopping is finite. This equation should be solved subject again to the boundary condition of e 1 on the exit boundary and e ± 0 on the complement of the exit. In summary, the bitting, or exit, probability coincides with the electrostatic potential when the boundary conditions of the diffusive and the electrostatic systems are the same.
A Guide to First-Passage Processes by Sidney Redner