Conditional mean first-passage time

Exercise. A Brownian particle obeys the diffusion equation (VIII.3.1) in the interval Lsplitting probabilities nL(X0) and (Xq) as functions of its starting point X0. Also the conditional mean first-passage times. 

The conditional mean first-passage time for those exits that take place between l and / + dl is... 

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... 

Exercise. For the random walk, nR m has been computed in (2.3). Find SRftn and subsequently the conditional mean first-passage time ... 

When L is not a reflecting boundary there is a conditional mean first-passage time My). It is obtained from the equation510 for the product quantity 3 (y) = nR(y)TR(y) ... 

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. 

Exercise. For the Smoluchowski equation in one dimension show that the conditional mean first-passage time r (y) obeys... 

Type (ii). The particle has a non-zero probability to reach the wall, but we show that it takes an infinite time to arrive. The conditional mean first-passage time at R — s is, according to (3.11),... 

Exercise. For the same case derive the expressions for the two conditional mean first-passage times tl(X0, V0) and tr X0, V0). 

For the solution of real tasks, depending on the concrete setup of the problem, either the forward or the backward Kolmogorov equation may be used. If the one-dimensional probability density with known initial distribution deserves needs to be determined, then it is natural to use the forward Kolmogorov equation. Contrariwise, if it is necessary to calculate the distribution of the mean first passage time as a function of initial state xo, then one should use the backward Kolmogorov equation. Let us now focus at the time on Eq. (2.6) as much widely used than (2.7) and discuss boundary conditions and methods of solution of this equation. 

Our derivation holds for L + 1 m < R — 2. One sees in the same way as before that (2.5) remains valid for m = R — 1 if one puts xRtR = 0. Physically this condition expresses the obvious fact that the mean first-passage time for a particle starting at the boundary R itself is zero. Also, a similar argument applied to m = L gives... 

This is identical with (2.5) because at the reflecting boundary rL = 0. The conclusion is When boundary L is reflecting the mean first-passage time at R is obtained by solving (2.5) for L m R — 1 with the boundary condition xr,r = 0- The solution is given in (2.10). 

If it happens that L is a reflecting boundary one has in analogy with (2.4) the boundary condition dnR/dy = 0 at L. It then follows trivially that nR(y) = 1 for all y, as expected. In this case one has a mean first-passage time Tfl(y). It obeys... 

The probability p x, t) that the absorbing state is reached between t and t + dt is readily computed from G x,t) as p = —dtG x,i). Note that p is already normalized due to the initial condition G(x, 0). Hence, the mean first passage time T x) equals... 

By following the time dependence of the reactant population and using the condition of detailed balance, Eq. (9), one can extract the desired rates. This methodology has been implemented successfully in a number of simulations see for example Ref. 43. Alternatively, one can compute the mean first passage time for all trajectories initiated at reactants and thus obtain the rate, cf. Ref. 44. 

A simpler way of finding the mean first passage time niij is to condition on the state after one step. Either we reach state j on the first step and Nij = 1, or we have used up a step and arrived in another state k and so a total of 1 + Nkj steps will be needed to reach state j. Hence... 

The probability distribution of the first passage time and its moments can be obtained from the Fokker-Planck equation for general situations of F x) and i/f (x), by following the standard procedures given in Section 6.6. In the present context, the average translocation time is the mean first passage time, which can be calculated by choosing the appropriate boundary conditions for P(x,t). 

The mean first passage time for the boundary condition BC2 follows from Equations 6.99, 6.100, and 6.102 as... 

Figure 5. Cell division as a first-passage time (FPT) problem, (a) Schematic of stochastic cell size increase from a common initial condition. Between times t and T + At, the some growth tracks cross the threshold size, 9. Using probability conservation, the cumulative probability that the size is greater than 9 (above the black dotted horizontal line) must be equal to the complement of the cumulative probability that the FPT is less than or equal to T Oeft of blue dotted vertical line at t). (b) Scahng of the FPT distribution. The shape of the mean-rescaled division time distribution is timescale invariant, that is, independent of k, when there is a single timescale, 1 /if ocr, in the FPT dynamics. (See insert for color representation of the figure.)...

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