Mean passage time

The mean passage time of 250 single erythrocytes and the number of rheological occlusions/250 erythrocytes is determined. 

The product J Eg)coQ is equal to Eg for a harmonic oscillator potential truncated at = Eg, and to 2Eg for a Morse potential with dissociation energy equal to Eg. Equation (2.41) is the low-friction limit result of Kramers. There are other methods to derive the results obtained in the previous section. One is to look for the eigenvalue with smallest positive real part of the ojjerator L defined so that dP/dt = — LP is the relevant Fokker-Planck or Smoluchowski equation. Under the usual condition of time scale separation this smallest real part is the escapie rate for a single well potential. Another way uses the concept of mean passage time. For the one-dimensional Fokker-Planck equation of the form... 

The pressure drop across the downstream filter as a function of the mean passage time of fluid element flowing from the upstream to the downstream filter displays a relaxation behaviour, figure 4a 4c. The measurements were made at several Reynolds numbers. With increasing Reynolds number the degradation of the polymer in the upstream filter increased, but the relaxation time and relaxation behaviour were unchanged. 

It was shown that the mean-passage time r splits naturally into 3 terms as follows... 

Fig. 7. Mean passage time for jumping across the unstable branch of Fig. 4 starting from the 1ow-temperature steady-state, as a function of the control parameter 6. Parameter values e = 0.08 ...

For a given value of j a solution of (15) is also an extremum of p(n.,j) for variable n-j thus the extrema of p(n , n2) are the solutions of (15) and of the symmetrical equation obtained by exchanging i and j. These equations may be treated numerically, and the results agree completely with the conclusions of the deterministic analysis (Fig.l), As for the mean passage times, computed from (11) they do not differ very much from, those of the previous sin >le approximation. 

To unbind from a protein the ligand has to move from a, the minimum of the potential U x), to 6, the maximum of U x). The mean first passage time t F) of such motion is (Izrailev et ah, 1997)... 

In this regime the applied force completely overwhelms the binding potential and the ligand is subject to free diffusion. The mean free passage time in this regime is equal to Td and is on the order of 25 ns. 

Calculation of Mean First Passage Times from Differential Recurrence Relations... 

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. 

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. 

We also remark that Eq. (5.44) may be decomposed into separate sets of equations for the odd and even ap(t) which are decoupled from each other. Essentially similar differential recurrence relations for a variety of relaxation problems may be derived as described in Refs. 4, 36, and 73-76, where the frequency response and correlation times were determined exactly using scalar or matrix continued fraction methods. Our purpose now is to demonstrate how such differential recurrence relations may be used to calculate mean first passage times by referring to the particular case of Eq. (5.44). 

The use of the differential recurrence relations to calculate the mean first passage time is based on the observation that if in Eq. (5.48) one ignores the term sY(x, s) (which is tantamount to assuming that the process is quasi-stationary, i.e., all characteristic frequencies associated with it are very small), then one has... 

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.

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. 

The standard theories of chemical kinetics are equilibrium theories in which a Maxwell-Boltzmann distribution of reactants is postulated to persist during a reaction.68 The equilibrium theory first passage time is the TV -> oo limit in Eq. (6), Corrections to it then are to be expected when the second term in this equation is no longer negligible, i.e., when N is not much greater than e — e- )-1. The mean first passage time and rate of activation deviate from their equilibrium value by more than 10% when... 

Using Eq. (8) for P(t), mean first passage times may then be calculated and compared to either the equilibrium result (N- oo) to test the range of validity of this assumption or to experiment. The agreement with experiment is not very good3 4 but then again the model outlined here is only a first step. The obvious refinement is the use of a more realistic... 

One may ask the following question. Suppose the random walker starts out at site m at t = 0 how long does it take him to reach a given site R for the first time This first-passage time is, of course, different for the different realizations of his walk and is therefore a random quantity. Our purpose is to find its probability distribution, and in particular the average or mean first-passage time ]... 

If this is equal to unity, then fR,m(t) is the probability density for the first-passage time at R. In that case the mean first-passage time is [compare (VI.7.5)]... 

Exercise. Write the corresponding equations for the case of a left exit point L < m. Exercise. Solve the first-passage problem for the simple symmetric random walk. Show that any site R is reached with probability nR m= 1, but that the mean first-passage time is infinite. 

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. 

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