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Dynamics [14] and has been very successful. Although these algorithms do not all require detailed balance and are successful for treating non-equilibrium steady states, application to general non-stationary dynamics is still ex- ploratory, and is so far only shown for WE [12] (although the authors step back from actually claiming this, pre- ferring to state that the method covers a ‘broad class of stochastic processes’ rather than the full range of stochas- tic non-stationary dynamics). The requirement for sta- tionarity in all existing methods apart from WE arises from the assumption that microstates from a given com- partment can be treated interchangably at the compart- ment boundaries - regardless of, often importantly, the duration of the path which has led to a given state. 

We generalise this strategy of compartmentation to non-steady-state systems; in essence only by fixing the duration of each trajectory fragment (here called a ‘shot’), so that the time evolution can easily be tracked. The resulting method, once sampling and reweight- ing schemes have been developed around this central premise, is termed Stochastic Process Rare Event Sam- pling (S-PRES). The most important design choice is the procedure used to ensure dynamically adaptive sampling rates for the different bins. This is achieved here using a variant of Rosenbluth sampling, as is sometimes used in FFS; rather than by moving the bins as has been investi- gated for WE. The choice to keep the bin positions fixed has the benefit of allowing a high-level and mathemati- cally friendly description of the dynamics to be developed online in the form of a time-dependent matrix of transi- tion frequencies between the bins. 

## II. S-PRES: ALGORITHM DESCRIPTION 

We define a scalar-valued coordinate λ as a function over the state space of our system. This coordinate is discretized into bins labelled by an index i . (Note that the phase space of the system need neither be discretized nor finite. These conditions only need to hold for the co- ordinate binning.) As an example choice for λ , one might use the number of particles in a liquid droplet forming in supersaturated vapour. 

The main goal of S-PRES is to observe a roughly con- stant number of forward transitions from each bin i on each interval [ t, t + τ ]; where a forward transition from i at t is defined as any shot where the microstate at time t + τ falls within a bin j > i . In this way unlikely transi- tions can be explored and sampled with high statistical accuracy. 

In principle, λ could be a vector instead of a scalar. The restriction to a scalar is used here to simplify the discussion. If using a vector-valued ⃗ λ , the concept of ‘forward’ becomes non-obvious. An example approach in two or more dimensions is to define a Hamming distance as the number of bin boundaries which remain to be be 

## A. Overview 

crossed in order to reach some target bin: ‘forward’ then describes any shot which decreases this Hamming dis- tance. 

## B. Importance Sampling for Adaptivity 

We carry out a variable number n t i of ‘shots’ (short dy- namics runs) of a fixed duration τ from the configurations in each bin i at each time t . (Here and in the following we use the term “configuration” for a microstate of the system at a given time on a given path. Two paths can, in principle, reach identical microstates at the same time. In this case the algorithm would still hold two configura- tions.) 

As we are considering stochastic processes, shots from the same starting configuration can be made to diverge by varying the random number seed used to generate the dynamics. n t i is adapted during the simulation to improve statistics. During the course of the simulation, the number of bins which are populated by configura- tions gradually increases (indicated by the triangles and squares in fig. 1) until transitions are sampled at each timestep from all bins which have a non-zero occupa- tion probability. In order to achieve a roughly constant number of forward transitions from each bin i , we select n t i based on the estimated transition probabilities at the previous timestep. The number of shots from bin i at time t + τ is defined as: 

where N is the target number of forward transitions, γ is a damping factor, R is the number of shots which moved forward from bin i to bins over higher ranges of λ on the step t − τ to t , (or R = 1 if this number is zero). The brackets ⌈ ⌉ indicate the ceiling function. This adap- tive sampling method makes S-PRES akin of the class of variational approaches to steady-state importance sam- pling (IS) described elsewhere [15]. Selection of values for the parameters N and γ is discussed in sec. II E. 

## C. Explanation of Sampling for Path Generation 

In order to enhance the exploration of rare states we apply a version of the pruned-enriched Rosenbluth method (PERM) [16], i.e. when picking configurations from a given bin as starting points for new shots, we do not select them with equal probabilities, but with a vari- able statistical weight which depends on the sampling history. In conventional PERM paths at a given time- point are either discarded, or selected for exactly one or two copies to extend to the next timepoint, based on lower and upper thresholds in their weights. The im- plementation presented here avoids having to set these thresholds. The number of branches n t i is shared out 

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