Splitting probability

The total probabilities to arrive at either one will be called the splitting probabilities... 

Exercise. If there is a reflecting boundary at some site Lm = 1. Exercise. For the asymmetric random walk one may take R = + oo, L = — oo. Show that the splitting probability uphill is zero and downhill equal to unity. 

Exercise. A Brownian particle obeys the diffusion equation (VIII.3.1) in the interval Lstarting point X0. Also the conditional mean first-passage times. 

Similarly one sees that for the splitting probabilities and exit times in the case (1.9) one merely has to solve... 

Consider again the general one-step process (1.1). We first establish an equation for the splitting probability nR m, that is, the probability that R is reached before L, after starting from a point m between both. Whenever L + 2 R — 2 the particle first has to jump to m + 1 or to m — 1, see... 

Thirdly we take a one-step process with two exits L and R. There are two splitting probabilities nR m and nLtm obeying (2.2) with the appropriate boundary conditions for each. There are also two conditional mean first-passage times xRm, iL m. In order to compute them we introduce the products R,m = nR,m R,m and L,m = L,m L,m- In the same way as before one argues that... 

Instead of the one-step process we now consider the generalized diffusion or Smoluchowski equation (1.9). Take a finite interval Ldifferential equation with the adjoint operator... 

Exercise. A particle obeys the ordinary diffusion equation in the space between two concentric spheres. Find the splitting probability and the conditional mean first-passage times. 

Take an interval (L, R) and an initial value y in it. We want to know the splitting probabilities and the time distribution for leaving the interval through either L or R. The probability Ol v. t) for being still in (L, R) without having made a jump across an end point obeys for L < y < R... 

The less ambitious approach in 2 through the adjoint equation aims only at the splitting probabilities and the mean first-passage times. We construct an equation analogous to (2.2) and (3.1). If at time t the value of Y equals x, it will have at t + At another value x with probability At JT(x x), or it has the same value x. The probability 7rK(x) that Y, starting at x, will exit through R obeys therefore the identity... 

To compute it we formulate the problem as one of splitting probability either the population dies out during its initial stages or it survives to grow to a macroscopic size. We therefore erect a boundary at some macroscopic value n = a, for which one may take ns and ask for the probabilities na and 7Zq that n will reach a or 0 first. The answer is given by (XII.2.8), or, in our present situation... 

Doublet splitting probably representing the sum of two proton splittings discussion of conformation. 

We can make the concept of the quality of a reaction coordinate q r) more precise by considering the so-called commitment probability, or conunittor. The conunit-tor pb (r) is defined as the probability that a trajectory started at configuration r with random momenta reaches state B before it reaches state A (see Fig. 10). (The commitment probability for state A is defined analogously.) The commitment probability was introduced as splitting probability already by Onsager, who used this concept to analyze ion pair recombination [262]. It has proven very useful in theoretical studies of protein folding, where the committor is known as pfou [263], and even in experimental work on liquid-solid nucleation [264]. Calculation of the probability pb r) involves a MaxweU-Boltzmann average over momentum space... 

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