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A second restriction, which should not in general be used but which was imposed on the FFS-like “flux” vari- ant of the method for the calculations carried out here, is to instantly terminate any paths which return to A ; and then re-initialise them with a random configuration from within A . This setup describes a slightly unphysical situation, but was required here to achieve exact corre- spondence of the definition of flux with existing steady- state FFS calculations [18], such that only paths from A to B which make the journey in a single pass without any return to A are considered. 

If it is preferred to calculate the time series of the state vector ⃗ d t and the matrices M t (or some other extracted observable) without specific definition of a flux then no special bins A or B are defined. 

## J. Non-Requirement for Poisson Statistics 

The assumption of Poisson statistics; that rare events occur independently and without correlation; is required by most existing methods [19]. This assumption may be an unwelcome limitation and is not required by S-PRES. (Although in the example of (sec. III) boundary condi- tions were set up so as to force Poisson behaviour). The weighted directed acyclic graph (WDAG) of configura- tions which S-PRES generates can be used to measure the deviation from Poisson statistics. Define P ( X$_{t}$ ′ | X$_{t}$ ) as the estimated conditional probability of event X at time t ′ > t given that the system also experienced X at time t . This is measured by performing a sum over the weights of configurations at t $^{′}$, counting only that set which are descended directly from configurations which experienced X at time t , { w$_{i}$ } , and a second sum over only those which experienced the event at both times, { w ′ i } : 

In this example use of the WDAG, the size of the set of joint events { w ′ i } may in practice be so small as to cause sampling errors unless the deviation from Poisson statis- tics is large or a large calculation is carried out. Storage of the entire WDAG is likely to be cumbersome for many practical applications in which the storage capacity to describe every configuration will quickly mount up. 

## III. APPLICATION TO RARE EVENTS IN THE ISING MODEL - GLAUBER-ISING UNDER IMPOSED SHEAR FLOW 

The example of nucleation in the 2D Ising model on a square lattice under imposed shear flow is a case that has 

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## A. Introduction to System 

been studied (although only for constant shear rate) us- ing FFS [10, 18]; allowing for direct comparison of our re- sults with those from an established method. This simple model exhibits rich non-equilibrium phase behaviour [20], however in this instance it is employed only to demon- strate the use of the sampling algorithm. 

The system was set up as follows (duplicating the FFS studies): Glauber dynamics were used to evolve the spins at each lattice site, meaning that at each ‘sweep’ L × L sites were chosen randomly to have their spins reassigned according to a Boltzmann-weighted probability. The sys- tem was prepared with all (65 × 65) spins down. A weak upward external field was applied, rendering the prepared state metastable relative to the stable state in which all or most spins are up. Shear flow can either accelerate or retard the formation of a nucleus of up-spins and the transition to the stable state, depending on the shear flow rate [10]. 

Shear flow was applied L times at each sweep by ran- domly selecting one of the L horizontal lines between rows of lattice sites and applying a move with probabil- ity ˙ γ such that all sites above it are translated one space to the left, with periodic wrapping such that the spin at i = 1 becomes the spin at i = L ; this is a simple model of infinite two-dimensional laminar Couette flow. 

A subtlety enters in the treatment of the vertical peri- odic boundaries (the interaction between sites with j = 1 and j = L ): in order to avoid shearing along this line un- less it has been explicitly selected, an offset pointer is maintained so that even after the spins in the row with j = 1 have been moved (after a shear at say, the bound- ary between j = 5 and j = 6) they remain in contact via periodic imaging with the same spins in the row j = N as they were before. 

A long discussion of the detailed implementation of this model system is available [18]. The notation ˙ γ to indicate the rate of the imposed shear flow is used here for consistency with this earlier work and has no relation to the γ of eqn. 1, which is used to indicate the ‘damping’ constant applied to stabilise sampling rates. 

In order to generalise the steady-state model having constant shear flow to a simple and directly comparable non-steady-state case we subjected the system to a time- series of three different shear flow rates within the low- shear (nucleation-enhancing) regime, allowing the nucle- ation rate to relax to the steady-state value after each change of shear flow rate. In order to produce directly comparable data to the FFS studies, the time dependent forward flux for the system was calculated using the same system parameters described for the steady state calcu- lations in refs. [10, 18]; which is to say size L = 65, cou- pling constant J = 0 . 65 k$_{B}$T and external field strength h = 0 . 05 k$_{B}$T . At ˙ γ = 0 the system can be considered to be in a state of quasi-equilibrium where a meta-stable basin and stable basin are separated by a large free en- ergy barrier. Classical Nucleation Theory (CNT) gives the size of the barrier to nucleation as ≈ 22 k$_{B}$T for this regime [21], signifying that a rare-event technique 

