content_image/380900.jpg
is strongly recommended to extract meaningful statistics by simulation. 

## B. Definition of Coordinate Bins for Shear-Ising Calculation 

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 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. 

## IV. APPLICATION TO RARE EVENTS IN THE ISING MODEL - 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 

## C. Results for Shear-Ising Calculation 

## A. Introduction to System 

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 $_{(1 +}$√$_{2) [25].}$ 

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 

<img src='content_image/380900.jpg'>

## B. Definition of Coordinate Bins for Kawasaki-Ising after a Quench 

