7 0 5e-13 1e-12 1.5e-12 2e-12 3000 4000 5000 φ ( sweeps × L 2 ) − 1 Time (MC sweeps) is strongly recommended to extract meaningful statistics by simulation. B. Definition of Coordinate Bins for Shear-Ising Calculation FIG. 2. Example use of S-PRES: Nucleation in the 2D Ising model under shear. Solid horizontal lines: ref- erence steady state nucleation rates for each value of imposed shear [10] (dashed lines show the reported errorbars). Fluc- tuating trace (red online): S-PRES time-series as the shear is changed. The coordinate over configurations was defined simply as the total number of up spins present, λ = N$_{up}$ . The source bin, A , was defined as λ < 25 and the sink bin B was defined as λ > 2005 (as in the FFS calculation [18]). The intervening space on λ was divided into 990 equal increments. It might have been possible to make a more sympathetic definition of the intervening bins, such as by spacing them more closely together for smaller values of λ where the dynamics on λ is expected to be slow, however this crude binning was found to be effective. The sampling parameters γ = 0 . 5, τ = 10 and N = 100 were used. Occasional ‘dead-end’ configurations mani- fested, where λ was large due to multiple isolated clusters of spins rather than due to a single nucleus: a maximum n$_{i}$ threshold of 2000 was therefore set, in order to prevent eqn. 1 from diverging due to these instances. it is difficult to predict their rates of collision and growth or shrinkage and the evolution of the size distribution of clusters over time. Despite these difficulties a theory based on the iterative evolution of a population vector of clusters of different sizes, p$_{n}$ ( t ) is available from the literature [22], which has not until now received direct validation from simulation studies (although a closely re- lated approach has had the benefit of such scrutiny [23]). The existence of an untested theory for such a simple but important model system is an ideal opportunity to fur- ther demonstrate the S-PRES method while at the same time making a small contribution to the basic study of phase-change dynamics. The equilibrium thermodynamics of this model are well understood, as are the phase-change dynamics in both the nucleation-dominated (surface-energy limited) and spinodal decomposition-dominated (diffusion lim- ited) regimes [24]. We carried out an instantaneous quench from T = ∞ to T = 0 . 6 T$_{c}$ , which lies between these two regimes, for a system of 100 × 100 spins with a 0 . 1 concentration of up-spins. The coupling constant J was set to 1 k$_{B}$T . T = 0 . 6 T$_{c}$ was set to 1 . 36151, using the Onsager result of T$_{c}$ = 2 /ln $_{(1 +}$√$_{2) [25].}$ It was necessary to run the calculation for 2500 MC sweeps at the initial shear rate ˙ γ = 0 . 04 before all bins of the coordinate λ were populated, allowing meaningful statistics to be collected. Fig. 2 shows the flux against time as the shear rates were changed ( ˙ γ = 0 . 04, 0 . 02, and 0 . 0). Horizontal lines (black) indicate steady state FFS data from a separate research group [10]; the trace (red online) is the S-PRES results. After each change of shear rate the time-dependent flux relaxes to the known steady state value (actually the quasi-equilibrium value in the case ˙ γ = 0 . 0), validating the method. The trace is an average over 100 independent runs. C. Results for Shear-Ising Calculation IV. APPLICATION TO RARE EVENTS IN THE ISING MODEL - KAWASAKI-ISING AFTER A QUENCH A. Introduction to System B. Definition of Coordinate Bins for Kawasaki-Ising after a Quench Phase separation in the 2D Ising model af- ter a quench into the temperature region between the nucleation-dominated and spinodal decomposition- dominated regimes is a quintessential problem in non- equilbirium dynamics. Under Kawasaki dynamics (some- times called a ‘lattice gas’) the total magnetisation is conserved and time-evolution is controlled by diffusion of spins. The base timescale of the system is set by one MC sweep, equal to N$_{up}$ attempts to move a random up- spin. Because the diffusive and evaporative behaviour of spin clusters is determined by both their size and shape As the coordinate we chose λ = ∑ $_{c}$( n$_{c}$ − 1), where n$_{c}$ is the number of spins in cluster c and a cluster is defined as a connected group of up spins. This coordinate was chosen because it is simple to calculate and has a value of zero when all up spins are isolated, increasing after any collision between spins or clusters. The lowest bin was defined as λ ≤ 9 and the highest bin was defined as λ ≥ 91, with the intervening integer values assigned to a