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.
Read or Download A Guide to First-Passage Processes PDF
Similar stochastic modeling books
Offers an creation to the analytical points of the speculation of finite Markov chain blending occasions and explains its advancements. This ebook seems to be at a number of theorems and derives them in basic methods, illustrated with examples. It contains spectral, logarithmic Sobolev suggestions, the evolving set method, and problems with nonreversibility.
This monograph is a concise creation to the stochastic calculus of adaptations (also often called Malliavin calculus) for techniques with jumps. it's written for researchers and graduate scholars who're attracted to Malliavin calculus for bounce methods. during this e-book strategies "with jumps" contains either natural leap tactics and jump-diffusions.
Electromagnetic advanced media are synthetic fabrics that have an effect on the propagation of electromagnetic waves in striking methods no longer frequently visible in nature. as a result of their wide selection of significant purposes, those fabrics were intensely studied over the last twenty-five years, commonly from the views of physics and engineering.
The learn of M-matrices, their inverses and discrete capability concept is now a well-established a part of linear algebra and the speculation of Markov chains. the focus of this monograph is the so-called inverse M-matrix challenge, which asks for a characterization of nonnegative matrices whose inverses are M-matrices.
Additional resources for A Guide to First-Passage Processes
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.