8 single bin each. The sampling parameters γ = 0 . 5, τ = 10 and N = 100 were used. C. Results for Kawasaki-Ising Calculation In order to prepare initial states including unusually large clusters the system was prepared in the T = ∞ regime using some few hundreds of iterations of S-PRES in order to achieve statistics down into the range p$_{n}$ = 10 − 12 before applying the quench. The probability distribution of cluster sizes ⃗ p$_{n}$ is not directly available from the probability distribution of re- action coordinate bins ⃗ d ; instead it was required to cal- culate it as an average over all configurations generated at each timestep, weighted according to eqn. 4. In fig. 3 we show S-PRES results for ⃗ p$_{n}$ . There is good qualita- tive agreement with the theory, which is shown in fig. 3- inset. The results shown are averaged over 10 indepen- dent calculations; error bars are the estimated standard errors over the 10 values. Brute force calculation in the T = ∞ regime is very cheap due to the lack of interac- tions between spins at this temperature, therefore data at t = 0 , T = ∞ from a brute force calculation separate to the S-PRES calculation is also shown in fig. 3 (and also the standard result p$_{n}$ = e − n/ 2 [26]). The exist- ing quantitative results for the infinite temperature case highlight a potential source of problems caused by the er- ror behaviour of S-PRES - until convergence is achieved, probabilities of rare states are reported as zero; meaning that S-PRES will converge on the correct values from below. To comprehensively explore the applicability of the Mirold-Binder theory is not the aim of this work, and would require further data over a wide range of temper- atures and concentrations, however to provide numerical results for systems previously accessible only to theory is an example of the type of research into phase change dynamics which can be carried out using S-PRES. V. APPLICATION TO RARE EVENTS IN A TIME-DEPENDENT ASYMMETRIC EXCLUSION PROCESS A. Introduction to System An Asymmetric Exclusion Process (ASEP) is a sim- ple model for driven stochastic transport. Here we dis- cuss the “parallel-open” (p-o) ASEP, as has been charac- terised by Sch¨ utz [27]. In this model, particles are intro- duced at the origin with a probability α at each sweep; and removed from the right boundary with probability β . Between the two boundaries, a deterministic update rule allows particles to move from left to right providing that a vacancy exists. When α = β , the system becomes crit- ical, with a divergent correlation length. Particles queue In order to validate S-PRES against the quite tractable time-dependent properties of the ASEP, the system was initialised without any particles; and allowed to gradually approach the steady state, in analogy to the morning traffic along a busy road. The length L was set as 500 sites and the parameters α and β were both set as 0 . 01. 12,000 brute force calculations were run, each of duration 10 7 sweeps. A single S-PRES calculation was also set up, with λ defined as the number of particles, divided over B. Simulation Setup up at the right boundary of the system, forming a block with density (1 − β ); and the remainder of the system has fast-moving traffic with density α . The phase boundary moves stochastically in the critical state according to a random walk. FIG. 3. (a) Example use of S-PRES: Temperature quench of the 2D Ising model with conserved order parameter. Calculated time evolution of the domain-size distribution is compared with theory [22] (inset). The the- ory is well outside the error bars (which are invisible except for very small p$_{n}$ ), but does provides qualitative agreement in so far as reproducing the shapes of the four curves. (b) Special case t = 0 , T = ∞ . At T = ∞ a brute-force calcula- tion is easy, so is superimposed on the S-PRES data down to p$_{n}$ ≈ 10 − $^{8}$. The standard result p ( n ) = e − n/ 2 is also shown. Colour online.