First passage time evolution times

Figure 10. Evolution of the survival probability for the potential Tf.v) — ax2 - for3 for different values of noise intensity the dashed curve denoted as MFPT (mean first passage time) represents exponential approximation with MFPT substituted into the factor of exponent.

Obviously, if Plx, Z xo) is known we can compute the mean first passage time from Eq. (8.154). We can also find an equation for this function, by operating with backward evolution operator t (xo) on both sides ofEq. (8.154). Recall that when the operator L is time independent the backward equation (8.122) takes the form (cf. Eq. (8.123b)) 9P(x, Z xo)/9Z = T f (xo)T (x, Z xo) where D is the adjoint... 

We defer the proof to the appendix. The reason for presenting two methods for evaluating the mean first passage time is based on their diflierent scopes of applicability. If we were only interested in T, then equation (11.30) would be preferable because it requires less computation. However, we are limited to the first moment [8]. The advantage of the first approach is that we obtain any moment by one integration. Moreover, we have access to the time evolution of the escape process which allows for a more detailed analysis. 

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). 

The theory of first passage times enables one to take a master equation like eq. (15.12) that describes the evolution of a probability distribution P t) for a population, and derive from it an equation for the probability distribution of times T t) for reaching a population m at time t when we start out with n at time zero. For the Szabo model of eq. (15.12), we find... 

In a realistic simulation, one initiates trajectories from the reactant well, which are thermally distributed and follows the evolution in time of the population. If the phenomenological master equations are correct, then one may readily extract the rate constants from this time evolution. This procedure has been implemented successfully for example, in Refs. 93,94. Alternatively, one can compute the mean first passage time for all trajectories initiated at reactants and thus obtain the rate, cf. Ref 95. 

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. 

Passage time distributions and the mean first passage time provide a useful way for analyzing the time evolution of stochastic processes. An application to chemical reactions dominated by barrier crossing is given in Section 14.4.2 and Problem 14.3. 

Figure 16 A typical time evolution of native contacts plotted for a given trajectory i versus first passage time t,. Thick lines indicate periods of time when stable native contacts are formed. Stability means that once formed, a contact q remains intact until the native state is reached. A critical nucleus is searched for in conformations recorded at times [5 (see the text for a more detailed explanation of how 5 is chosen).

Fig. 5.21 Time evolution of (component) populations following the pump pulse. Populations of the two diabatic electronic states (Pi, P2), and contributions from the first two passages through the conical intersection (Pj, are shown. (Reprinted with permission from Y. Arasaki et al., J. Chem. Phys. 132, 124307 (2010)).

A first approximation to the description of the full process is to assume that, by some mechanism independent of the fluctuations, the system remains uniform in space. In this "zero-dimensional" description, considerable attention was focussed on the decay from an initial unstable state, and on the passage times between simultaneously stable states [25-29]. Two characteristic scales emerge from this analysis. For the evolution around the unstable state, the time needed for the probability distribution to forget the initial condition and begin to develop peaks toward the stable attractors is... 

Figure 10.9 (a) Sketch of a generic free energy profile for translocation as a function of the number of monomers that are already translocated into the receiver compartment, (b) Sketch of two trajectories for the time evolution of the number of monomers translocated into the receiver compartment. The translocation time is the first passage time. 

---