0e+00 1e+04 2e+04 0 50 100 150 200 250 Speedup Number of Bins ˙ γ = 0 . 04 ˙ γ = 0 . 02 ˙ γ = 0 . 00 0 0.2 0.4 0.6 0.8 1 0 100 200 300 < p B ( i, t ) > Bin i t=30 0 0 t=40 0 0 t=50 0 0 10 FIG. 6. Scaling of algorithm efficiency: performance is more robust to excessive numbers of phase space bins than it is to insufficient numbers of bins. As the probability of the rare event in consideration decreases (smaller φ ) the speedup becomes proportionally more. Each trace is an average over 10 independent calculations. Colour online. FIG. 5. Committor probabilities for 2D-Ising under shear , assuming that the system is initialised in bin i at each time t = 3000; 4000; 5000 (corresponding to ˙ γ = 0 . 04, 0 . 02, and 0 . 0). The committor p$_{B}$ ( i ) moves to higher bins for later timesteps (lower shears), which is consistent with the lower nucleation rates observed. Colour online. committor distribution, as discussed for the equilibrium 3D Ising model in ref. [31]. These statistics can be cal- culated with respect to λ as above; or with respect to an arbitrary variable by using the WDAG. numbers of equally spaced bins, even for small numbers of bins where the assumption of closely spaced bins no longer holds. Fig. 6 shows speedup versus the number of bins for calculations at the three different shear rates. At the shear rates for which nucleation is more rare, the speedup is proportionally greater. The speedup was ro- bust to the use of excessive numbers of bins. The smaller system of L = 50 was chosen for the benchmarking be- cause it is computationally less expensive; reaction fluxes were roughly equal to those observed for the larger sys- tem, with φ = 1 . 5 × 10 − $^{12}$, 0 . 8 × 10 − 12 and 0 . 3 × 10 − 12 for the three shear rates. If the S-PRES calculation is set up in order to find rare states, then it has two phases. In the first phase (‘popu- lation’) the goal is to achieve a state whereby one or more configurations are associated with each bin, allowing rare events to be observed. In the second phase (‘observa- tion’) the goal is to continue the dynamics and observe the time-evolving behaviour. To make a loose scaling argument from equilibrium statistical mechanics, if we assume that the number of bins N$_{B}$ is large enough that no significant free energy barriers exist between adjacent bins, but that only one new bin is populated at each iteration of the algorithm, then the total time required for the population phase should be roughly proportional to N 2 $_{B}$/ 2. During the observation phase, the time re- quired per iteration should continue to be linear with the number of bins. Therefore the population phase is considered as the performance bottleneck of the method and ‘speedup’ is defined as the expected number of MC sweeps needed to observe the first rare event using brute force (equal to L $^{2}$/φ for the shear-Ising example system) divided by the expected number of MC sweeps needed to observe the first rare event using S-PRES, which marks the completion of the population phase. We repeated the shear calculations of (sec. III) at con- stant ˙ γ = 0 . 04 , 0 . 02 and 0 . 0 and with L = 50; for various S-PRES is robust to this situation in the sense that S-shaped trajectories can be developed by the algorithm because paths which move backward as well as forward in λ are generated and stored; however it is still better to choose coordinate projections which are near-monotonic A common thought experiment used to test coordinate-based rare event methods such as FFS, umbrella sampling or S-PRES is to imagine a system for which the the projected coordinate λ is non-monotonic or sometimes orthogonal with respect to the true reac- tion coordinate [19, 32]. A practical example of this is protein folding, where even simple proteins and peptides can move through sequences of transition states which are dissimilar to each other and to both the unfolded and folded conformations [33, 34], making it difficult to define a useful projection of the progress of the reaction without detailed prior knowledge of the the folding mechanism. B. Robustness to Non-Monotonic Binning A. Scaling with Relation to Fineness of Binning VII. SCALING AND CHOICE OF BINNING