First passage time moment calculations

One can obtain an exact analytic solution to the first Pontryagin equation only in a few simple cases. That is why in practice one is restricted by the calculation of moments of the first passage time of absorbing boundaries, and, in particular, by the mean and the variance of the first passage time. 

When the initial probability distribution is not a delta function, but some arbitrary function Wo(xo) where xq C (c,d), then it is possible to calculate moments of the first passage time, averaged over initial probability distribution ... 

We have already noted the difference between the Langevin description of stochastic processes in terms of the stochastic variables, and the master or Fokker-Planck equations that focus on their probabilities. Still, these descriptions are equivalent to each other when applied to the same process and variables. It should be possible to extract information on the dynamics of stochastic variables from the time evolution of their probabihty distribution, for example, the Fokker-Planck equation. Here we show that this is indeed so by addressing the passage time distribution associated with a given stochastic process. In particular we will see (problem 14.3) that the first moment of this distribution, the mean first passage time, is very useful for calculating rates. 

Equation 13.4 is useful since the calculations of moments are easier than estimates of the distributions (the statistics required for accurate estimates of the moments are much smaller). We emphasize that the first passage time is not the inverse of a rate constant. The use of a rate constant assumes that the time evolution is exponential. The mean first passage time is a well-defined measure of the progress of the reaction with no reference to exponential relaxation. The inverse of the mean of the first passage time corresponds to a rate constant only if the distribution of first passage times is exponential. This is true not only for the overall mean first passage time bnt also for the local first passage times computed as the first moments of the distribntions K, .(i). 

FIGURE 13.6 The first passage times sampled from milestone 9. The positive number are from terminating trajectories that made it to milestone 10 and the negative times from the trajectories that terminate on milestone 8. A total of 99 events are shown as sticks. The statistic is insufficient to obtain accurate estimate of the density but allows for sound calculation of the first moments. The times are in femtoseconds. 

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). 

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