1e-08 1e-07 1e-06 1e-05 0.0001 0.001 1e+06 1e+07 P(full) Station ary L imi t Br u te F o rc e S-P R E S F ick’s La w 1e-50 1e-40 1e-30 1e-20 1e-10 1 100000 1e+06 1e+07 P(full) MC Sweeps S t a t i o n a r y Li m it B r u t e For c e S -PR E S F i c k ’s La w VI. EXAMPLE USE OF THE MATRICES M 9 The structure of M with respect to time can be anal- ysed in order to estimate the usefulness of the coordinate projection which has been employed and to pursue in- sight into the mechanics of the system under considera- tion. An example is the extraction of committor proba- bilities. The ‘committor probability’ p$_{B}$ ( s ) or the ultimate probability that a given configuration s will complete the reaction before returning to some initial state is a quan- tity generally of interest in the analysis of rare events. In a non-stationary system this value can change with respect to time. An easy estimate of committor proba- bilities at a given time t$_{0}$ can be achieved by using the time series of M in the following procedure: FIG. 4. Probability that the ASEP is completely full given that it is empty at t = 0. The S-PRES calculation agrees with theory over 46 decades. Colour online. C. Results for ASEP Calculation 100 equal-sized bins. The S-PRES parameters τ = 500 and γ = 0 . 5 were used. The procedure above gives the committor only with re- spect to the bins on the projected coordinate λ ; the main purpose of such an analysis is to evaluate the usefulness of the particular definition of λ . It is considered that the closest possible identity between λ and the committor probability gives the most efficiently enhanced sampling for rare event methods [28]. If λ does not determine p$_{B}$ or if the granularity of the binning is large near to sharp changes in p$_{B}$ , then S-PRES becomes less useful and alternative strategies might be required. If an initial rough calculation can be made to work, then it is possible to record online (without the assumption of mixing) the mapping between λ and some different observable using eqn. 4. In the case that large computer memory is avail- able then the entire WDAG of configurations connected by path segments can be stored; allowing formal methods of projection onto subspaces of manageable dimensional- ity [29, 30] to be experimented with offline. Fig. 5 shows the committor probability distributions for three timepoints in the evolution of the variable-shear system of (sec. III) corresponding to three different shear- rates. The shifting of the distribution to the right for lower shears is consistent with the smaller reaction flux observed. This effect is due to the fact that the fre- quency of cluster collisions decreases with decreasing ˙ γ more quickly than does the “cluster evaporation” rate. Further statistics of interest for a typical system might include the transmission coefficient κ or the width of the The S-PRES and brute force calculations converged to the steady-state limit and the S-PRES calculation was also able to confirm that kinetic properties were very ac- curately predicted by the assumption of Fickian diffusion of the phase boundary. This is shown in fig. 4. The rare event in this case is the full occupation of the system, i.e. the particle number equalling the num- ber of sites. Statistics were collected from the S-PRES and brute force runs on the probability of the rare event P ( full ). This is available in the steady-state limit from [27] as P ( full )$_{t}$$_{→∞}$ = 2(1 − α ) L/ $^{2}$/L . Assuming that the phase boundary moves as a random walk starting from the origin at t = 0, and that the density of the dense phase is constant; then the time-dependent value of P ( full ) is readily available by numerically iterating Fick’s first law, beginning with the probability density defined as 1 . 0 at the origin and 0 elsewhere. 1. Create two ‘sink’ bins A and B s.t. ∀ t > t$_{0}$ : M t A,A = 1, M t B,B = 1, M t A,j = A = 0, M t B,j = B = 0. 2. Repeat for each bin b / ∈ { A, B } (a) Initialise a vector ⃗ d s.t. d$_{b}$ = 1 and d$_{i}$ = 0 ∀ i = b . (b) Apply the time series of modified M t to each ⃗ d , beginning at t = t$_{0}$ until d$_{i}$ ≈ 0 ∀ i / ∈ { A, B } . (c) d$_{B}$ now holds the expected value of p$_{B}$ ( s ) over the bin b .