Sampling Rare Events in Non-Equilibrium and Non-Stationary Systems Joshua T. Berryman ∗ and Tanja Schilling Johannes Gutenberg Universit¨ at, Mainz, Germany and Universit´e du Luxembourg, Luxembourg Although many computational methods for rare event sampling exist, this type of calculation is not usually practical for general nonequilibrium conditions, with macroscopically irreversible dynamics and away from both stationary and metastable states. A novel method for calculating the time-series of the probability of a rare event is presented which is designed for these conditions. The method is validated for the cases of the Glauber-Ising model under time-varying shear flow, the Kawasaki- Ising model after a quench into the region between nucleation dominated and spinodal decomposition dominated phase change dynamics, and the parallel open asymmetric exclusion process (p-o ASEP). The method requires a subdivision of the phase space of the system: it is benchmarked and found to scale well for increasingly fine subdivisions, meaning that it can be applied without detailed foreknowledge of the physically important reaction pathways. I. INTRODUCTION A. The Rare Event Literature B. Phase-Space Binning and Reweighting Events which are highly improbable often have great importance to the behaviour of a system. The classic example of this is nucleation of a raindrop from super- saturated water vapour. Because droplets smaller than a critical radius are energetically unfavourable, the forma- tion of a super-critical droplet is unlikely to occur on the timescale of thermal motion of water molecules. Hence a straightforward computer simulation would waste much CPU time on unimportant fluctuations before producing an event of interest. analogues or extensions of this idea are rapidly being de- veloped; for example by considering the transition prob- abilities between states [7] rather than directly assigning a free energy and the associated Boltzmann probability distribution directly to states themselves. In order to numerically study rare events under con- ditions where the concept of a free energy landscape is problematic, Transition Path Sampling (TPS) [8], For- ward Flux Sampling (FFS) [9, 10], Weighted Ensemble (WE) [11] and a suite of related methods have been de- veloped. The basic idea of these methods is to selec- tively sample from the set of pathways which the system can take, by increasing the number of pathways in the important regions of the state space of the system, but compensating by attaching a variable statistical weight to each path. This group of methods is aimed at steady state non-equilibrium systems, and except for one very recent paper on WE [12] (published after this work was substantially complete) the potential for adapting or re- formulating them to give a time-dependent description of non-stationary dynamics has not yet been explored. Many processes (such as quenching, aging, ignition and impact) are naturally framed in a strictly non-steady or time-dependent way: beyond the equilibrium/quasi- equilibrium and also the stationary nonequilibrium treat- ments. The development of non-stationary rare event methods is therefore of potentially great importance. A large number of approaches have been developed to solve so-called “rare event” problems. Many of these ap- proaches are based on Transition State Theory [1, 2], i.e. on the concept of a quasi-equilibrium free energy landscape and of a particular “slow” motion of the sys- tem within this landscape. The landscape is imagined as consisting of basins linked together by ‘Transition Paths’ passing over saddle points. This landscape can be mapped using equilbrium methods including e.g. um- brella sampling [3], multi-canonical sampling [4] and Wang-Landau sampling [5]; and the motion across saddle points, once they are identified, can then be simulated di- rectly by initialising molecular dynamics simulations near to the saddle or even just reconstructed from the poten- tial of mean force [6]. This approach of proceeding from a free energy surface to an understanding of the kinetics is principled and highly attractive but is limited to systems for which such a surface can be meaningfully defined and practically computed. The starting point for this work is a phase space bin- ning according to some macroscopic coordinate λ (which is often called the “reaction coordinate” although it is not usually the true reaction coordinate of the process). Bi- ased sampling is then performed so as to generate paths which move through specific bins on λ . This strategy of projecting the phase space of the system onto some sub- space of one or more dimensions; dividing the subspace with a set of partitions and then running short trajec- tory paths in or between compartments is common to FFS [9], WE [11], Milestoning [13] and Boxed Molecular For systems away from equilibrium the concept of free energy becomes problematic, although theories which use ∗ josh.berryman@uni.lu arXiv:1001.2456v3 [cond-mat.mes-hall] 6 Nov 2010