THE JOURNAL OF CHEMICAL PHYSICS 148, 124120 (2018)

Importance sampling large deviations in nonequilibrium steady states. IUshnish Ray,1,a) Garnet Kin-Lic Chan,1,b) and David T. Limmer2,3,4,c)1Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena,California 91125, USA2Department of Chemistry, University of California, Berkeley, California 94609, USA3Kavli Energy NanoScience Institute, Berkeley, California 94609, USA4Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94609, USA

(Received 3 September 2017; accepted 5 March 2018; published online 29 March 2018)

Large deviation functions contain information on the stability and response of systems driven intononequilibrium steady states and in such a way are similar to free energies for systems at equilib-rium. As with equilibrium free energies, evaluating large deviation functions numerically for all butthe simplest systems is difficult because by construction they depend on exponentially rare events.In this first paper of a series, we evaluate different trajectory-based sampling methods capable ofcomputing large deviation functions of time integrated observables within nonequilibrium steadystates. We illustrate some convergence criteria and best practices using a number of different models,including a biased Brownian walker, a driven lattice gas, and a model of self-assembly. We showhow two popular methods for sampling trajectory ensembles, transition path sampling and diffusionMonte Carlo, suffer from exponentially diverging correlations in trajectory space as a function ofthe bias parameter when estimating large deviation functions. Improving the efficiencies of thesealgorithms requires introducing guiding functions for the trajectories. Published by AIP Publishing.https://doi.org/10.1063/1.5003151

I. INTRODUCTION

Free energy calculations play a central role in molecu-lar simulations of chemical, biological, and material systems.As the governing function for stability, free energies are aprinciple quantity of statistical physics. With the exception ofspecial cases, the evaluation of free energies requires numer-ical computation. The development and ease of implementa-tion of methods such as umbrella sampling, metadynamics,and Wang-Landau sampling1–3 has made the computation ofequilibrium free energies textbook material.4 With these andrelated techniques, free energy calculations of realistic mod-els of matter are routinely used to compute phase diagrams,to infer thermodynamic response, and to estimate rates of rareevents.5–7 However, such calculations are traditionally limitedto instances where systems are at thermodynamic equilibrium,as most algorithms are typically formulated with an implicitreliance on an underlying Boltzmann distribution of configura-tions. Large deviation theory provides a theoretical frameworkfor extending free energy calculations to systems away fromequilibrium, with large deviation functions acting as the ana-log of free energies for such systems.8 Though not identicalto a thermodynamic potential, large deviation functions never-theless encode stability and often response of nonequilibriumsystems. While methods exist to evaluate large deviation func-tions numerically,9–13 few studies on their performance14,15

or extensions have been reported.16 Here we evaluate

a)Electronic mail: uray@caltech.edub)Electronic mail: garnetc@caltech.educ)Electronic mail: dlimmer@berkeley.edu

the convergence of two methods, transition path sampling(TPS)17 and diffusion Monte Carlo (DMC).18 We find thateach samples the same distribution of trajectories and can inprinciple converge large deviation functions. However, bothmethods suffer from sampling deficiencies that make appli-cations to high-dimensional systems difficult. By illustratingthese properties, we provide a basis for what is necessary tosystematically extend them for use in molecular models ofnonequilibrium steady states.

Large deviation theory underlies much of the recentprogress in nonequilibrium statistical mechanics. Within itscontext, strikingly simple and general properties for systemsdriven away from thermal equilibrium have been derived.These include relationships between far-from-equilibriumaverages and equilibrium free energies, such as the Jarzyn-ski equality and Crooks fluctuation theorem, and symme-tries within nonequilibrium ensembles, such as the Lebowitz-Spohn and Gallovoti-Cohen symmetries.19–22 Large deviationfunctions are cumulant generating functions for the fluctua-tions of quantities in arbitrary physical systems. Many physicalsystems of fundamental and applied interest operate awayfrom thermal equilibrium, either because boundary conditionsgenerate a flux of material or energy, or because the indi-vidual constituents take in energy to move or exert force.Therefore computing large deviation functions as a means ofcharacterizing the likelihood of fluctuations about a nonequi-librium state has the potential to broaden our understand-ing of physical processes ranging from crystal growth andmolecular self-assembly, to nanoscale fluid flows and insta-bilities, to the organization of living systems and other activematter.23–29

0021-9606/2018/148(12)/124120/15/$30.00 148, 124120-1 Published by AIP Publishing.

124120-2 Ray, Chan, and Limmer J. Chem. Phys. 148, 124120 (2018)

Large deviation functions are averages taken over atrajectory ensemble. As such, to compute large deviationfunctions for high-dimensional systems, methods for sam-pling trajectory ensembles must be constructed and employed.Beginning with transition path sampling,30 followed by itsgeneralizations, such as transition interface sampling31 andthe finite temperature string method,32 such tools have beendeveloped. As will be elaborated on below, such methodssample trajectory space by a sequential random update, result-ing in a Monte Carlo walk that uniformly samples trajec-tories. Similarly, methods that advance an ensemble of tra-jectories in parallel, such as forward flux sampling33 andthe cloning algorithm,18 have been established. These meth-ods also sample a trajectory ensemble though they do soby directly representing the ensemble by a collection oftrajectories that are simultaneously updated in time. Suchmethods also sample a trajectory ensemble uniformly and,as with sequential update Monte Carlo methods, admit theaddition of importance sampling to compute large deviationfunctions.

Overwhelmingly, these methods have been applied tosample rare trajectories evolved with detailed balance dynam-ics.34 Notably, such studies have found much success in thestudy of the glass transition35–37 and protein folding.29,38 Therehave been some efforts to extend trajectory based importancesampling to driven systems, but these have been largely con-fined to the calculation of simpler observables, rather thanlarge deviation functions.39–41 The few applications of pathsampling methods to compute large deviations have beenlimited either to processes occurring with moderately highprobability within the nonequilibrium steady state or to low-dimensional model systems.42–44 This is because the statisti-cal efficiency of the standard algorithms becomes exponen-tially small with the exponentially rare fluctuations neededto be sampled. This has made the computational effort toconverge large deviation functions intractable for many real-istic systems. As will be illustrated below, the origin of thelow sampling efficiency is the small overlap between thedistribution of trajectories generated with a model’s dynam-ics and the distribution whose dynamical fluctuations mostcontribute to the large deviation function at a given bias.This difficulty manifests itself differently in various meth-ods, with either exponentially small acceptance probabilityor exponentially large variance in the weights. The gener-ality points to a need to develop robust importance sam-pling schemes on top of these trajectory based Monte Carloprocedures.

In the following, we outline the basic theory of largedeviations of physical processes and how they are estimatedstochastically with path sampling. This serves to clarify howdisparate methods like transition path sampling and diffusionMonte Carlo sample the same trajectory spaces in essentiallythe same way. We then explore the convergence of standardimplementations of different methods by studying systems ofincreasing complexity: a biased Brownian particle on a peri-odic potential, a 1d driven lattice gas with volume exclusion,and a 2d model of self-assembly. We use these models to illu-minate convergence criteria and the origin of statistical ineffi-ciencies. Throughout we focus on the commonalities between

two canonical Monte Carlo methods and aim to draw gen-eral conclusions about their efficiencies and how they can beextended to complex systems. These lessons serve to motivatethe formulation of a hierarchy of methods to mitigate the lowsampling efficiencies by introducing an auxiliary dynamics.The formalism follows the technique of using a trial wave-function in diffusion Monte Carlo and is discussed in Paper IIof this work.45

II. THEORY AND MONTE CARLO ALGORITHMS

For compactness and ease of notation, we focus onlarge deviation functions for processes described by time-independent, Markovian stochastic dynamics. While exten-sions of large deviation theory to deterministic, non-Markovian dynamics, and time-dependent processes exist,these generalizations will not be considered here. Similarly,we will only consider ensembles of trajectories with a fixedobservation time though generalizations to a fixed number ofevents and fluctuating observation times are straightforward.46

Much of the theory of large deviations summarized here isexpanded upon in Ref. 8.

In general, Markovian processes obey a continuity equa-tion of the form

@t ⇢(Ct , t) =W⇢(Ct , t), (1)

where ⇢(Ct , t) is the probability of observing a configurationof the system, C, at a time t and W is the linear operator in thatconfigurational space that propagates trajectories. If C spans adiscrete space, then W is a transition rate matrix and Eq. (1)is a master equation. If C spans a continuous space, then Wis a Liouvillian and Eq. (1) is a generalized Fokker-Planckequation. In either case, provided W is irreducible, Eq. (1)generates a unique steady state in the long time limit, which ingeneral produces non-vanishing currents and whose configu-rations do not follow Boltzmann statistics or some generaliza-tion. Only in instances where W obeys detailed balance is thesteady state an equilibrium distribution in which all currentsvanish.47

For general nonequilibrium steady states, there exist twogeneric classes of time extensive observables: configurationtype observables that are sums of functions of individualconfigurations,

Oa =

tNX

t=1

oa(Ct) , (2)

where oa is an arbitrary function of a single configuration, andcurrent type observables that are sums of functions of changesin configurations,

Ob =

tNX

t=1

ob(Ct�1 ! Ct) , (3)

where ob is an arbitrary function of configurations at adjacenttimes. Within a nonequilibrium steady state, either observ-able type has an associated probability distribution that can beconstructed as a marginal distribution over an ensemble of tra-jectories. For example, for an observable that is a combinationof configurational and current types, O = Oa + Ob,

124120-3 Ray, Chan, and Limmer J. Chem. Phys. 148, 124120 (2018)

p[O] = h� 266664O � tNX

t=0

oa(Ct) +tNX

t=1

ob(Ct�1 ! Ct)377775 i

=X

C(tN )

P[C(tN )]�266664O �

tNX

t=0

oa(Ct) +tNX

t=1

ob(Ct�1 ! Ct)377775 ,

(4)where h. . .i denotes the path ensemble average within thesteady state distribution generated by the system dynamicsand �(x) is Dirac’s delta function. A member of the trajectoryensemble is denoted C(tN ) = {C0, C1, . . . , CtN }, which is a vec-tor of all of the configurations that a system has visited over anobservation time, tN . Generically the probability for observinga given trajectory, C(tN ), is given by

P[C(tN )] = ⇢(C0)tNY

t=1

u(Ct�1 ! Ct) , (5)

where ⇢(C0) represents the distribution of initial conditions,the suppression of an explicit time argument denotes a sta-tionary distribution, and u(. . .) are the transition probabilities.For a given nonequilibrium steady state, ⇢(C0) is the stationarydistribution and the transition probabilities are encoded in thepropagator, W.

Observables that are correlated over a finite amount oftime admit an asymptotic, time intensive form, for theirdistribution function,

t�1N ln p[O] ⇠ ��(O/tN ) , (6)

where �(O/tN ) is the rate function or generalized entropy ands denotes the long time limit that is an equality up to sub-extensive terms /1/tN . In this limit, a generating function canbe computed by taking the Laplace transform of Eq. (4),

he��Oi =X

C(tN )

P[C(tN )]e��O ⇠ e (�)tN , (7)

where (�) is known as the large deviation function or gen-eralized free energy and � is a counting field conjugate to theobservable. Like an equilibrium free energy, the large devia-tion function is a cumulant generating function, and by takingderivatives with respect to �,

@n (�)/@�n���=0 = (�1)n 1tNh(�O)ni , (8)

the nth scaled cumulant of O is generated. Unlike in equi-librium, in general � is not related to a physical field, liketemperature. When the asymptotic limits exists and the ratefunction is a smooth convex function, then the rate functionand large deviation function are related through a Legendretransform,

(�) = infO

[�(O/tN ) + �O/tN ], (9)

where inf is the largest lower bound taken over all O. Thisstructure is reminiscent of thermodynamics, where Legendretransforms change ensembles between conjugate variables.Like in equilibrium thermodynamics, when the large deviationfunction is non-analytic, or nonconvex, this transform losesinformation about the fluctuations of the system, as it willreturn the convex hull of �(O/tN ). Away from these points,when the rate function is convex, the long time limit ensures

an equivalence of ensembles48 between those conditioned ona value of O and those with fixed field �, resulting in anaverage value of O as in Eq. (8). This is analogous to equilib-rium in the thermodynamic limit away from points of phasetransitions.49

As an exponential average, estimating large deviationfunctions through straightforward simulations is difficult forlarge �. Stochastic algorithms have been designed to solvethis problem by means of Monte Carlo sampling. In equi-librium free energy calculations, difficulties associated withexponentially rare fluctuations are overcome by introducing asampling bias in the form of additional terms in the Hamil-tonian. Provided the additional terms, the system can be sim-ulated or sampled and the bias can be corrected for easilybecause the distribution function is the known Boltzmannform. This basic procedure is the basis of numerous methodssuch as umbrella sampling, metadynamics, and Wang-Landausampling.1–3

However, equilibrium importance sampling methods donot translate straightforwardly to nonequilibrium steady states,as the distribution of configurations is not known generically.Added biases in the form of terms in the Hamiltonian, or prop-agator, are not easily reweighted. However, provided a meansof sampling trajectories uniformly from their steady state dis-tribution, added biases in the evolution of the system can beconstructed to push the system into rare regions and theireffects exactly corrected in order to return unbiased estimatesof those rare fluctuations. To importance sample large devia-tion functions in nonequilibrium steady states, the trajectoryensemble is biased, or tilted, to the form

P�[C(tN )] = P[C(tN )]e��O[C(tN )]� (�)tN , (10)

where the large deviation function (�) is the normalizationconstant computable as in Eq. (9). The new distribution oftrajectories, P�[C(tN )], represent those that contribute most tothe large deviation function at a given value of �. Ensembleaverages for arbitrary observables, O0, within the unbiaseddistribution and the tilted one, denoted h. . .i�, are relatedby

hO0[C(tN )]i� =hO0[C(tN )]e��O[C(tN )]ihe��O[C(tN )]i , (11)

where the denominator is exp[ (�)tN ]. In any stochastic sam-pling method, the ensemble average is estimated by a finite sumof trajectories. Differences between methods dictate how theensemble of trajectories is constructed, and their efficienciesare determined by how well correlations between trajectoriescan be engineered and how easily biases or other constraintscan be added and removed. In this manner, the importancesampling is not that different from equilibrium free energycalculations, with the exception that the simplifying forms tocorrect for the added bias provided by the Boltzmann distribu-tion are replaced by implicit forms that must be computedalong each trajectory. Below we discuss two Monte Carlomethods for sampling trajectories from nonequilibrium steadystates with the correct weights, to which additional impor-tance sampling can be added in order to efficiently evaluatelarge deviation and rate functions.

124120-4 Ray, Chan, and Limmer J. Chem. Phys. 148, 124120 (2018)

A. Transition path sampling

The distribution defined by Eq. (5) can be sampled usinga Monte Carlo algorithm, analogous to path integral quan-tum Monte Carlo. Chandler and co-workers first identifiedthis analogy and developed transition path sampling (TPS)to importance sample dynamical events occurring in genericclassical systems.30 Since its inception, TPS has become animportant tool for investigating rare events, typically of theform of barrier crossings in systems evolving around an equi-librium.17 In recent years, it has been used to compute largedeviation functions for systems evolving with detailed bal-ance preserving dynamics. Most notably, this has been donefor model glass-formers35,37,50 and glassy polymers.29,38 Onlya very few studies have employed it in the computation oflarge deviation functions for driven systems, or nonequilib-rium steady states.40,43 Nonequilibrium systems constrain thetypes of Monte Carlo moves that can be easily accommodated,due to incomplete knowledge of the steady state distribution.Below we review the basics of TPS and how it can be used tocompute large deviation functions.

TPS generates an unbiased ensemble of trajectories by asequential random update to an existing trajectory, using a vari-ety of different update moves and accepting or rejecting eachmove according to a Metropolis rule. Usually, the update rulesare collective and result in the simultaneous change of manyconfigurations along the trajectory. To derive the Metropolisacceptance rules, the targeted stationary distribution must beknown. In order to sample exponentially rare fluctuations ofan observable, O, and obtain a reasonable estimate of the largedeviation function, trajectories are weighted in proportion tothe path ensemble defined in Eq. (10). Given this distribution,the acceptance criterion for updating the MC walk betweentwo trajectories Co(tN ) and Cn(tN ) can be determined by thesteady state criterion

P�[Co]�[Co ! Cn]Ac[Co ! Cn]

= P�[Cn]�[Cn ! Co]Ac[Cn ! Co], (12)

where Ac[. . .] is the acceptance probability and �[. . .] is thegeneration probability and we have suppressed the argumentsof the trajectories, assuming that they are all tN in length. Rear-ranging, we can put the acceptance probability in a Metropolisform

Ac[Co ! Cn] = Min⇣1, e��(O[Cn]�O[Co])�⌦⌘ , (13)

where the first term proportional to � is the bias away fromtypical events and

⌦ = � lnP[Cn]�[Cn ! Co]P[Co]�[Co ! Cn]

(14)

depends on how new trajectories are generated. If they aregenerated using physical dynamics, independent of �, then ⌦is a measure of the entropy production to transform between thetwo trajectories.51 The specific form of the entropy productiondepends on the details of the Monte Carlo move.

For a so-called forward shooting move, a point is picked atrandom along the trajectory, Ct⇤ , and the system is reintegrated.

FIG. 1. Illustration of trajectory update moves used in TPS: (a) forwardshooting, (b) backward shooting, and (c) forward shifting.

This is illustrated in Fig. 1(a). The acceptance can be computedby expanding Eq. (14),

e�⌦=

QtNt=t⇤+1 u(Cn

t�1!Cnt )⇣g[Cn

t⇤ !Cot⇤ ]QtN

t=t⇤+1 u[Cot�1!Co

t ]⌘

QtNt=t⇤+1 u[Co

t�1!Cot ]⇣g[Co

t⇤ !Cnt⇤ ]QtN

t=t⇤+1 u[Cnt�1!Cn

t ]⌘

=g[Cn

t⇤ !Cot⇤ ]

g[Cot⇤ ! Cn

t⇤ ]= 1, (15)

where transition probabilities up to t⇤ trivially cancel in thenumerator and denominator because those pieces of the tra-jectory are the same. The last equality follows from assumingthat the shooting point is changed symmetrically with prob-ability g[Cn

t⇤ ! Cot⇤ ], independent of the configuration, for

example, by drawing new random noise for the stochasticintegration. With such a procedure, the move is accepted inproportion to the ratio of exp(��O) between the new and oldtrajectory.

In order to have an ergodic Markov Chain, the entire tra-jectory from the initial condition must be sampled. Unlike inequilibrium, for dynamics that do not obey detailed balancethis is problematic, as in general the acceptance probability forbackward moves cannot be manipulated to yield unity even for� = 0. For a backward shooting move, illustrated in Fig. 1(b),the acceptance criterion is given by

e�⌦ =

⇣⇢(Cn

0)Qt⇤

t=1 u[Cnt�1 ! Cn

t ]⌘ ⇣QtN

t=t⇤+1 u[Cot�1 ! Co

t ]⌘

⇣⇢(Co

0)Qt⇤

t=1 u[Cot�1 ! Co

t+1]⌘ ⇣QtN

t=t⇤+1 u[Cnt�1 ! Cn

t ]⌘ ,

(16)

where u[. . .] is the time-reverse transition rate and Ci is the cor-responding time reversed state of the system. From stochasticthermodynamics,20,51 the time reverse rates are related to theforward rates by the entropy produced, ⌅,

u[Cnt�1 ! Cn

t ]

u[Cnt�1 ! Cn

t ]= e⌅

n(t�1,t) . (17)

Substituting this relation into the acceptance probability, itis clear that the acceptance rate is exponentially small andcontains two terms: a time extensive entropy production and aboundary term with information on the initial conditions,

124120-5 Ray, Chan, and Limmer J. Chem. Phys. 148, 124120 (2018)

e�⌦ = e�[Wn(t0,t⇤)�Wo(t0,t⇤)] , (18)

where

Wo,n(t0, t⇤) =t⇤X

t=1

⌅o,n(t � 1, t) � ln ⇢(Co,n0 ) . (19)

This ratio is problematic because the relative weights for theinitial distribution are often not known. Even in cases wherethe initial distribution is known, or can be neglected as in a longtime limit, for a nonequilibrium steady state, the time extensivepart means that backward shooting moves are exponentiallyunlikely to be accepted, with a rate proportional to the lengthof the move and the number of driven degrees of freedom inthe configuration space.

Alternatively, the initial condition and early time segmentsof the trajectory can be updated using forward shifting moves,illustrated in Fig. 1(c). In such a move, a configuration fromthe middle of the trajectory is randomly selected, Ct⇤ , andmoved to the beginning of a new proposed trajectory. In orderto conserve the total length of the trajectory, an additionalsegment is generated from the last configuration of the old tra-jectory. The acceptance criterion for such a move can be writtenas

e�⌦ =⇢(Cn

t⇤ )Qt⇤

t=1 u[Cnt ! Cn

t�1]

⇢(Co0)Qt⇤

t=1 u[Cot�1 ! Co

t ]= 1 , (20)

where the final equality assumes that the nonequilibriumdriving is time-reversal symmetric and that the system isevolving within a stationary steady state. This cancelationoccurs because under these assumptions the proposal and pathweights are generated with the same dynamics. This is anal-ogous to previous work whereby new initial conditions aregenerated by short trajectories, followed by reintegration ofthe full path.39 Because the dynamics we consider are stochas-tic and irreducible, forward shooting and forward shifting aresufficient to ensure an ergodic Markov chain, as over repeatedTPS moves, the entire trajectory is replaced. This is distinctfrom other forward only propagating methods like forwardflux sampling where the initial conditions are not updated,which can result in biased sampling if sufficiently large initialdistributions are not used.34,52 However, trajectories decorre-late more slowly without backward shooting as many shootingand shifting moves are needed to generate completely newtrajectories. The acceptance criterion for backward shiftingcan be analogously derived and presents similar problems asbackward shooting.

The moves outlined above, together with the acceptancecriterion of Eq. (13), allow TPS to be used to sample trajec-tory ensembles of arbitrary nonequilibrium systems in pro-portion to their correct statistical weight. Expectation valuesof the form of Eq. (11) can be estimated straightforwardlyby

hO0i� = 1Ntraj

NtrajX

i=1

O0[Ci(tN )|�], (21)

where N traj is the number of trajectories harvested by TPS.This is because by construction they sample P�[C(tN )] withouta bias. Such expectation values can be related to the unbiased

steady state distribution using Eq. (11). In order to estimatethe large deviation or rate function, simulations can then berun with multiple values of �, and provided the distributionscollected at various values of � overlap, (�) can be computedfrom the relation

��(O) = �(O) + �O/tN + (�), (22)

where ��(O) is the marginal distribution, or unnormalizedrate function, generated from Eq. (10). For multiple distri-butions, calculations with many values of � can be related tothe standard histogram reweighting techniques, such as theweighted histogram analysis method or the multistate Bennetacceptance ratio (MBAR), which provide optimal estimatesof (�), and unbiased �(O) computed far into the tails of itsdistribution.53,54 Histogram reweighting techniques can onlydetermine (�) up to a global constant but because the unbi-ased dynamics are normalized, (0) = 0, so the full functionof (�) can be uniquely determined. These procedures area straightforward generalization of traditional umbrella sam-pling calculations in equilibrium, only now with an ensembleof trajectories.

B. Di�usion Monte Carlo

An alternative to sampling trajectory space with sequen-tial random updates is to sample trajectories in parallel. Thisis the strategy proposed in diffusion Monte Carlo (DMC)and has been adopted for the study of nonequilibrium clas-sical stochastic dynamics by algorithms known as forwardflux sampling and the so-called cloning algorithm.12,16,55 Inboth methods, an ensemble of initial conditions is generatedusing some prior distribution. Each initial condition, referredto here as a walker, is propagated for some short time, cre-ating an ensemble of short trajectories. Depending on thespecific algorithm and observable of interest, importance sam-pling is incorporated by assigning weights to each walker thataccumulate over the short trajectories and upon iterations ofparallel updates to the full trajectory ensemble. Generically,walkers are weighted in proportion to the biased ensembledefined in Eq. (10). The cloning algorithm offsets the expo-nential weights with a population dynamics, whereby walkerswith low weight are deleted and those with high weight arereplicated or “cloned.” The cloning algorithm in particularhas been successful in computing large deviation functionsfor model lattice systems56,57 and some small atomistic sys-tems.58 Mostly it has been used to study models of trans-port in 1 dimension.18,59,60 Below we review the basics ofDMC and how it can be used to compute large deviationfunctions.

The form of P�[C(t)] suggests an alternative way in whichtrajectories may be generated. Expanding P�, we get

P�[C(tN )] =⇢(C0)

QtNt=1 u(Ct�1 ! Ct)e��o(Ct�1!Ct )

PC(tN ) ⇢(C0)

QtNt=1 u(Ct�1 ! Ct)e��o(Ct�1!Ct )

,

(23)where we have written O as a current type observable,O[C(tN )] =

PtNt=1 o(Ct�1 ! Ct), otherwise it could be an addi-

tive function of single configurations. The argument insidethe product is the transition probability times a bias factor.

124120-6 Ray, Chan, and Limmer J. Chem. Phys. 148, 124120 (2018)

This combination of probabilities does not represent a physicaldynamics, as it is unnormalized. However, it can be interpretedas a population dynamics as was done by Giardina, Kurchan,and Peliti and is done in the standard DMC, where the noncon-servative part proportional to the bias is represented by addingand removing walkers.11

In particular, in the cloning algorithm, trajectories arepropagated in two steps. First Nw walkers are propa-gated according to the normalized dynamics specified byu[Ct�1! Ct] for a time, �t. Over this time, the bias isaccumulated according to

wi(t,�t) = exp (��O[Ci(t � �t), Ci(t)]) , (24)

where O[Ct��t , Ct] =Pt

t0=t��t o[Ct0 ! Ct0+1] is the collectedcurrent, which due to the multiplicative structure of the Markovchain is simply summed in the exponential. After the trajec-tory integration, ni(t) identical copies of the ith trajectory aregenerated in proportion to wi(t,�t). This is known as a branch-ing step. In the standard quantum-DMC, �t is chosen to bethe time step corresponding to trotterization of the Hamilto-nian.61 Additionally, instead of using the bias to branch, it isadvantageous to incorporate it into the integration step and usethe normalization as the branching weight. However, in manyapplications of non-equilibrium stationary states, calculatingthe normalization may not be straightforward.

There are different ways of performing the branching stepthat either holds the total number of walkers constant or not. Ina simple formulation with a non-constant walker population,the number of replicated walkers is determined stochasticallyby

ni(t) = bwi(t,�t) + ⇠c (25)

where ⇠ is a uniform random number between 0 and 1 andb. . .c is the floor function. Alternatively, the floor functionmay be avoided by using a partial weight for the walker, whichdecreases the statistical noise. The process of integration fol-lowed by branching is continued until t = tN . In this case,the large deviation function can be directly evaluated from thewalker population as

(�) =1tN

lntN/�tY

t=1

Nw(t)Nw(t � 1)

, (26)

which follows from the equivalence between the normalizationconstant of the biased distribution and the large deviation func-tion. This relation clarifies that the number of walkers neededto resolve (�) is exponentially large in �, and the time exten-sive observable, O. Because of this exponential dependence,this method is not practical and requires that the population iscontrolled in some way to mitigate walker explosion or totalwalker deletion.

One way to do population control is to formulate the DMCalgorithm with a constant walker population. In this case, asbefore each walker is initially integrated over a time�t. At thisend of the short trajectory, ni(t) copies of the ith walker aregenerated stochastically by

ni(t) =6666664Nw

wi(t,�t)PNw

j=1 wi(t,�t)+ ⇠

7777775, (27)

where ⇠ is a uniform random number between 0 and 1. Gener-ally this process will result in a different population of walkers,and so deficiencies or excesses in the walker population arecompensated by cloning or deleting walkers uniformly. In thiscase, the large deviation function can be evaluated at each timeas

t(�) = ln1

Nw

NwX

i=1

wi(t), (28)

which is an exponential average over the bias factors of eachwalker. This local estimate can be improved by averaging overthe observation time,

(�) =1tN

tN/�tX

t=1

t(�), (29)

which upon repeated cycles of integration and populationdynamics yields a refined estimate of (�). Since the pop-ulation dynamics represents the path probability, averages canbe computed with walker weights equal to

hO0i� = 1Nw

NwX

i=1

O0[Ci(tN )], (30)

where the trajectory C(t) has been generated according toP�[C(t)] in Eq. (23). This estimator is only valid if the num-ber of walkers is held constant. We note here that instead ofusing a constant population of walkers, it is possible to use anon-constant walker population with a chemical potential asis done in the standard quantum DMC. This can lead to bet-ter statistics since in this case the uniform redistribution ofexcesses or deficit of walkers is avoided.62

The cloning algorithm utilizes the multiplicative structureof the Markov dynamics to tune the relative times betweenpopulation steps, �t, in order to effect adjustments in the sizeof the biases used to add or remove trajectories. The choiceof �t is critical as it scales the overlap between the exponen-tial bias and the generating dynamics. Population sizes can bedecreased for smaller �t since the distribution from the gen-erating dynamics is wider for smaller �t and the exponentialdecays more slowly since O is time-extensive. However, thewider the distribution, the larger the variance in t(�), so localestimates of the trajectory distribution and its bias are poorer.This tension results in a sensitivity to �t that requires it to bechosen such that it is large enough to affect good local esti-mates of the large deviation function, but small enough thatthe overlap between the bias factor and the bare dynamics islarge. However, it cannot be too large since population controlis only done after �t, and so the systematic error from finitepopulation will grow with �t as will the stochastic error fromfinite walker population.

The time-series that result from DMC can be consideredin two ways: in terms of continuous or discontinuous tra-jectories as illustrated in Fig. 2. The continuous trajectoriescan be constructed by tracing the configurations backwardsfrom C(tN ) keeping those branches that have been cloned. Inthis case, trajectories sample exactly P�[C(t)] and expecta-tion values can be computed just as in TPS. This means thathistogram reweighing procedures outlined in Sec. II A canbe used to compute �(O), instead of the typical method of

124120-7 Ray, Chan, and Limmer J. Chem. Phys. 148, 124120 (2018)

FIG. 2. Illustration of the diffusion Monte Carlo algorithm. The red andblack lines represent different walker trajectories. (a) Illustrates the multi-plicative weight of trajectories constructed during forward propagation. (b)Non-constant walker population method, whereby branching and deletion isdone in proportion towi. (c) Constant walker population method where branch-ing and deletion result in continuous (solid lines) and discontinuous (dashedlines) trajectories.

computing �(O) from an inverse of the Legendre transform.The discontinuous paths correspond to trajectories constructedby tracking the configurations of a walker as they are propa-gated forward in time. In this case, each cloning event intro-duces an abrupt change in configuration space since a walker’sconfiguration may be replaced by configurations from otherwalkers that carry larger weights. The ensemble described bydiscontinuous trajectories is not equal to the trajectories of thebiased ensemble generally. For sufficiently large number ofwalkers and long enough propagation time, the two ensemblesare strongly overlapping.

C. Comparisons between TPS and DMC

Both TPS and DMC are Monte Carlo procedures to sam-ple trajectory space with the correct path weight. While theirimplementations and executions are different, both evaluateexpectation values stochastically and do so by updating thesystem of interest using its unmodified propagator. In bothmethods, the computational cost scales as the number of trajec-tories generated times the observation time. Systematic errorsarising from finite length trajectories or finite walker popula-tions can be alleviated in both, as will be shown below. Theresulting Monte Carlo walk in trajectory space is explored dif-ferently in the different algorithms. However, provided eachcan relax to a stationary distribution, estimates of comparablestatistical accuracy require the same number of uncorrelatedtrajectories and thus the same computational effort.

In TPS, the sequential update means that complete trajec-tories of length tN are constructed and so global constraintssuch as temporal boundary conditions imposed on the trajec-tory ensembles are naturally accommodated. Indeed, transitionpath sampling was conceived of as a means of sampling equi-librium trajectory ensembles with hard constraints on the ini-tial and final configurations, as needed in the study of barriercrossing events. However, because large deviation functionsresult from a long time limit, individual long trajectories must

be constructed. Maintaining correlations for such long trajec-tories with the standard shooting and shifting moves is difficultespecially when the ensemble sampled by the natural systemdynamics does not have significant overlap with the ensemblethat contributes to the large deviation function at finite �. Aswill be discussed below, this can result in significant statisticalerrors.

In contrast to TPS, DMC constructs paths in parallel usingan ensemble of walkers and implicitly benefits from moderncomputer architectures. However, the propagation in DMC isuni-directional with path construction relying heavily on thedistribution of paths in a local interval of time, �t. Accuratesampling of the local distribution therefore becomes exceed-ingly important and due to the exponential scaling of the biasbecomes hard to accomplish with limited walker populations.Formally the number of walkers needed to accurately con-verges the calculation scales exponentially with �. This com-plexity results in individual trajectories carrying exponentiallylarger weights than others, which means they are likely to bereplicated more often. This can produce an ensemble of walk-ers that becomes highly correlated over long times and finitewalker populations. Modulating these walker correlations isthe key to controlling statistical errors in DMC.

III. CONVERGENCE CRITERIA AND STATISTICALUNCERTAINTY

In order to understand the utility of these algorithms, andundercover their limitations, we have studied three differentmodels. The first is forced Brownian motion that evolves incontinuous space and time, but whose large deviation functioncan be computed exactly. The second is a driven lattice gasthat evolves in continuous time but discrete space, whose two-body interactions result in correlated many-body dynamics.The third is a minimal model of nonequilibrium self-assemblythat evolves in discrete space and time but exhibits a nontriv-ial phase diagram including critical fluctuations. These threesystems allow us to ground a discussion of the systematicand statistical errors that occur in calculating large deviationfunctions.

A. Long time or large population limit

Finite time and finite walker population14 are the largestsource of systematic error in the calculation of large devia-tion functions. Other sources of error that exist independentof importance sampling, such as those resulting from the dis-cretization of continuous time dynamics are well understoodand controllable.6 The definitions for the limiting forms ofthe probability distribution or its cumulant generating func-tion, Eqs. (6) and (9), are exponential equalities. Finite timeeffects result from sub-exponential terms whose origins arefrom finite time correlations or from temporal boundary con-ditions. In TPS, the long time limit is approached by increasingtN directly. Deviations from the asymptotic values of � or come directly from the temporal boundary conditions that arefree to fluctuate independently and will bias initial and finalparts of the trajectory toward values typical of the system at� = 0. In DMC, provided trajectories have been integratedinto the steady state, the trajectory space is rotated into a

124120-8 Ray, Chan, and Limmer J. Chem. Phys. 148, 124120 (2018)

space of walkers, and the effective long time limit can beapproached by increasing walker population at fixed integra-tion time. Because of the assumed ergodicity of the nonequi-librium steady state, averages computed from an ensemble oftrajectories are expected to be equivalent to those computedfrom an infinitely long single trajectory. In DMC, providedthe fixed integration time is long compared to the correlationtimes of the system dynamics, � or can be converged solelyby increasing the walker population. Alternatively, one cantake a hybrid approach, where a finite number of walkers canbe compensated by longer propagation times.

To explore the systematic errors arising in the calculationsof large deviation functions, we consider a paradigmatic modelfor driven systems: Brownian motion in a one-dimensionalperiodic potential with an external force.63,64 The system issketched in Fig. 3(a). This model has been studied extensivelyin the context of the fluctuation theorem and is exactly solvablenumerically by diagonalization of the Fokker-Planck operatorthrough basis set expansion techniques.65 As such, it providesan ideal reference to understand errors resulting from finitetrajectory length or finite walker population.

The overdamped dynamics of a Brownian particle movingin a closed ring geometry with circumference L and exter-nal periodic potential V (x) = vo cos (2⇡x/L) obeys a Langevinequation

x = �rV (x) + f + ⇠ , (31)

where the dot denotes the derivative with respect to t. The parti-cle is driven out of equilibrium through the non-gradient forcef, resulting in a non-vanishing current around the ring. The

FIG. 3. Rare entropy production fluctuations of a driven Brownian walker.(a) Typical trajectories for instances of large f ⇤ = 1.5 (red) and small f ⇤ = 0.5(blue). Inset is a schematic of the model. (b) Rate functions for the entropyproduction for f ⇤ = 0.8 (triangles), 1.0 (squares), and 1.5 (circles) computedwith TPS (red) and DMC (blue). The black curves are numerically exactresults.

random force ⇠ describes the effective interactions betweenthe particle and environment fixed at temperature T withstatistics

h⇠(t)i = 0 h⇠(t)⇠(t 0)i = 2µT�(t � t 0), (32)

where the mobility of the particle has been set to unity. In theresults we present below, we set L = T = 1, vo = 2 and considerthe effects of changing a non-dimensionalized external force,f ⇤ = f /2⇡vo � 0.

The dynamics generated by Eq. (32) are well character-ized by two limiting behaviors. In the limit that f ⇤ is small,a typical trajectory is localized near a potential minimumwith rare excursions to adjacent minima. Under such condi-tions, the average particle flux around the ring is small. Inthe limit that f ⇤ is large, a typical trajectory exhibits frequenthops between adjacent minima, biased in the direction of thenon-gradient force. Under these conditions, the average par-ticle flux around the ring is large. Examples of both types oftrajectories are shown in Fig. 3(a). A parameter that charac-terizes the dynamical behavior of the system is the entropyproduction

⌃(t) =⌅ t

0f dx(t 0) = f�x(t) (33)

which is time extensive and in this case simply proportional tothe particle flux around the ring. We denote its time intensivecounterpart, �(tN ) = ⌃(tN )/tN . In order to evaluate the finitetime effects for TPS and DMC, we compute the rate functionfor entropy production, �(�), and its associated large deviationfunction, (�), for this model.

To simulate the model of driven Brownian motion, we usea second-order stochastic Runga-Kutta integrator with timestep 0.001.66 For calculations using TPS, we simulated trajec-tories of length tN = 2000, using forward shooting and shiftingmoves and correlated noises.51 Trajectories are harvested usingthe acceptance criterion given in Eq. (13), with a bias of exp[ �tN�(t)], for � values ranging between 1 and 1 in incre-ments of 0.025. Shooting and shifting points were optimizedto result in a 30% acceptance ratio and around 106 trajecto-ries were harvested for each value of �. Rate functions werecomputed using MBAR and error bars were estimated usinga bootstrapping analysis. For calculations using DMC, we set�t = 0.05 and tN = 20, and a constant walker population ofNw = 10 000. Branching probabilities were computed fromEq. (27). Values of � between 1 and 1 were used to compute (�) using Eq. (29), and a numerical Legendre transform wasused to determine �(�).

Figure 3(b) compares the rate functions for three differentcases, characterized by an increasing force, f ⇤. For finite f ⇤,�(�) exhibits a minimum at a finite value for the entropy pro-duction. With increasing f ⇤, the mean value of� increases andthe curves broaden. In addition, there is an asymmetry whichbecomes more notable with increasing f ⇤. This latter featurearises from breaking microscopic reversibility and is codifiedin the fluctuation theorem,

�(�) � �(��) = � . (34)

As derived for diffusion processes by Kurchan,67 the fluctu-ation theorem relates the probability of generating entropy to

124120-9 Ray, Chan, and Limmer J. Chem. Phys. 148, 124120 (2018)

the probability of destroying it. Equation (34) implies that asthe entropy production gets larger, the likelihood of seeing afluctuation that reduces entropy becomes exponentially rarer.Figure 3(b) compares �(�) computed with TPS and DMCto numerically exact results. For all three cases studied, bothmethods are able to recover the exact result quantitatively.From the fluctuation theorem, Eq. (34), it is clear that trajec-tories that generate negative entropy are rarer, and as a result,the error bars for both TPS and DMC are larger for � < 0.The ability to reproduce the exact result itself is not surprisingfor this simple model. There are no approximations introducedin the algorithms, so in the limit of well sampled ensembles,the converged expectation values should agree with the exactresult within their statistical error bar. It is then only a questionof how quickly the long time or large walker population limitis reached so that finite time effects are converged to a givenaccuracy.

For the model of driven Brownian motion, we find thatthe long time limit for the rate function is approached as

�(�) � �(�, tN ) / 1/tN + O(1/t2N ), (35)

where �(�, tN ) is the finite time estimate of the asymptotic�(�). Therefore to leading order, finite time corrections scaleas 1/tN with a proportionality constant that depends on thecorrelation time for fluctuations in�. Data taken from TPS cal-culations with varying tN are shown in Fig. 4(a) for a numberof different � values. In this model, time correlations arise dueto barrier crossing events that result in large changes in � rel-ative to the typical small amplitude changes from fluctuationsof the particle at the bottom of the potential well. For smallf ⇤ or � > 0, these barrier crossing events occur infrequently,

FIG. 4. Systematic errors in large deviation function of the entropy produc-tion for the driven Brownian walker with f ⇤ = 0.8, vo = 2. (a) Finite time errorcomputed with TPS for � = 0.25 (blue), 0.0 (red), and 0.25 (black). Solidlines are best fits of the form A/tN . (b) Finite walker number error computedwith DMC for � = 0.5 (blue), 0.2 (red), and 0.2 (black). Solid lines are bestfits of the form B/Nw.

and as a consequence, the times along a trajectory where theyoccur are uncorrelated. By contrast, for large f ⇤ or � < 0,barrier crossings occur often, and accordingly � fluctuationsexhibit longer correlation times. Because of this, the prefactoraccompanying the 1/tN correction to the rate function becomeslarger for increasing �, or increasing f ⇤, as shown in Fig. 4(a).

Provided�t is large enough to produce the steady state, thesystematic errors in DMC depend on the number of walkers.Formally, for branching rates obeying Eq. (27), the numberof walkers required to offset the bias is exponential in ���t.For a constant walker number algorithm, Nemoto et al. haveshown68 that the leading order error in the rate function scalesas

(�) � (�, Nw) / 1/Nw + O(1/N2w), (36)

where (�) is the Legendre transform of �(�) and (�, Nw),its finite population estimate. This is indeed what we find forthe driven Brownian motion model. As shown in Fig. 4(b), forvarious values of �, we find that systematic errors in the esti-mate of the large deviation function are inversely proportionalto the walker population. This is due to the exponential error inrepresenting the steady state distribution made by DMC with afinite number of walkers. As an expectation value of an expo-nential quantity, (�) consequently depends linearly on thispopulation error. The coefficient determining the proportion-ality between the stochastic error and the number of walkersis related to the number of correlated walkers, which increasewith increasing �.

B. Correlations in trajectory space

In TPS and DMC with branching, expectation values areestimated by summing over trajectories generated with the cor-rect path weights so that each entry into the sum is weightedequally. Because in practice this sum is finite, the existence ofstatistical errors determines the accuracy with which expecta-tion values, like large deviation functions, can be computed.For a given number of trajectories, the statistical efficiencyis determined by the number of trajectories that are uncor-related. In TPS, the sequential update results in correlationsbetween trajectories that originate from a common trajectoryand share some configurations in common. The relevant mea-sure of these correlations is the time, in number of TPS moves,over which trajectories remain correlated. In DMC, the paral-lel branching rules replace lower weight walkers with thoseof higher weight, resulting in a given walker history that canbe traced back to some number of common ancestors. Cor-relations within the trajectory ensemble are measured by thenumber of walkers that originate from the same source. Inboth cases, for traditional implementations of these algorithms,we find that correlations rapidly increase for increasingly rarefluctuations. In practice, this renders the convergence of largedeviation functions untenable for all but the smallest systemsize, or �.

To explore the origin of statistical uncertainties in theestimation of large deviation functions, we consider a fam-ily of models known as simple exclusion processes (SEPs).These models are well studied examples of one dimensionaltransport and are considered paradigmatic models of nonequi-

124120-10 Ray, Chan, and Limmer J. Chem. Phys. 148, 124120 (2018)

librium many-body physics.69 The simple exclusion processevolves on a lattice with L sites, and in the following, we con-sider the open case whereby particles are injected or removedat site 0 and L 1. At a given time, t, the configurationof the system is defined by a set of occupation numbers, ni

= {0 1}, e.g., C(t) = {0, 1, . . . , 1, 1}. The probability of observ-ing a given configuration at a specified time, ⇢(Ct), can befound from the solution of the continuity equation in Eq. (1),with a Markovian rate matrix, W, whose elements have theform

W(C, C0) = �r(C) +X

C,C0w(C, C0), (37)

where r(C) =P

C w(C, C0) is the exit rate and w(C, C0) has theelements w = wL +

Pi wi + wR, where

wL =

266664�↵ �

↵ ��

377775 , wR =

266664�� �

� ��

377775 ,

wi =

26666666664

0 0 0 0

0 �q p 0

0 q �p 0

0 0 0 0

37777777775.

The matrices wL and wR act on the first and last sites, respec-tively, in a basis of single particle occupancy, i.e., ni = {0 1}.Particles are inserted at the boundaries with rates ↵ and � andremoved with rates � and �. The matrices wi act on all L sitesand are expressed in a multi-particle basis, i.e., nini+1 = {0001 10 11}. Particles move to the right with rate p and to theleft with rate q, subject to the constraint of single site occu-pancy. This hard core constraint results in correlations betweenparticles moving on the lattice. These rates and the hard coreexclusion are illustrated in Fig. 5(a).

For most values of ↵, �, �, �, q, and p, an average currentof particles flows through the system. We denote the particlecurrent by Q(tN ), which is equal to the number of hops to theright minus the number of hops to the left over a time tN ,

Q(tN ) =tNX

t=1

L�1X

i=0

�i+1(t)�i(t � 1) � �i(t)�i+1(t � 1), (38)

where �i is the Kronecker delta function for site i and the sumruns over the lattice and tN . Finite steady state currents can begenerated either by boundary driven or bulk driven processes.For p = q, boundary driven currents occur when ↵, �, and�, � are such that they establish mean densities on the leftand right ends of the lattice that are unequal. For q , p, bulkdriven currents are produced and remain finite in the infinitelattice size limit, unlike boundary driven currents which decayfollowing Fick’s law as 1/L. For q , p and different valuesof the boundary rates, the model exhibits a nontrivial phasediagram, with different density profiles and mean currents.70

A representative trajectory in the maximum current phase70

is shown in Fig. 5(a). As a relatively low dimensional many-body system, SEP models represent an ideal testing ground forstudying statistical errors in the computation of large deviationfunctions.

We have computed large deviation functions for parame-ters that include both bulk and boundary driven transport. For

FIG. 5. Large deviation functions for the particle current in the SEP mod-els. (a) Schematic of the SEP model, 1d lattice gas with nearest neighborhopping rates and double occupancy exclusion. A typical trajectory shown inspace-time, where the dark lattice sites denote an occupied position. (b) Largedeviation function for L = 8 boundary (circles) and bulk (squares) driven SEPmodels, computed with TPS (red), DMC (blue), and exact diagonalization(solid line).

the case of boundary driven transport, we study a model with↵ = � = 0.9, � = � = 0.1, and q = p = 0.5. For the case of bulkdriven transport, we study a model with ↵ = � = � = � = 0.5,and p = 0.75 and q = 0.25. For both cases, we consider fairlysmall lattices, L = 8. The dynamics are propagated by samplingthe trotterized rate matrix using kinetic Monte Carlo,71 with atime step of 0.05 that is commensurate with single particlemoves. For calculations using TPS, we simulated trajecto-ries of length tN = 20, using forward shooting and shiftingmoves. Trajectories are harvested using the acceptance crite-rion given in Eq. (13), with the bias exp[ �Q(tN )]. Shootingand shifting points were optimized to result in a 30% accep-tance ratio and around 105–106 trajectories were harvested foreach value of �. Large deviation functions were solved self-consistently with MBAR as in Eq. (22). For calculations usingDMC, we used integration times,�t = 0.5, tN = 100, and a con-stant walker population of Nw = 2000. Branching probabilitieswere computed from Eq. (27), and (�) was estimated fromEq. (29).

Figure 5(b) show the large deviation functions computedfor fluctuations of Q. As before, there is quantitative agree-ment between both TPS and DMC. Figure 5(b) compares theresults for both bulk and boundary driven cases obtained bysampling with TPS and DMC and exact numerical resultsobtained by computing the maximum eigenvalue of the many-body rate matrix using exact diagonalization. Across the rangeof �’s, the sampled values agree with those numerically exactresults. In all cases, the simple convex shape of the large devi-ation functions is a consequence of the fluctuation theorem,the one derived by Lebowitz and Spohn for Markov jumpprocesses. For a 1d system such as SEP, the total entropy

124120-11 Ray, Chan, and Limmer J. Chem. Phys. 148, 124120 (2018)

production along a trajectory is linearly proportional to thecurrent.72 The proportionality constant is known as the affinity,A = ln(↵�pL 1/��qL 1)/L + 1, and results in a slightly differentsymmetry for current fluctuations,

(�) = (A � �), (39)

where for an equilibrium system, A = 0, and the fluctuation the-orem is again a statement of microscopic reversibility.21 ForSEP models, the value of A depends on whether the currentis generated by bulk or boundary driving, and as a conse-quence, the curves in Fig. 5(b) are symmetric about differentvalues.

As before, the main systematic errors in converging (�)scale as 1/tN for TPS and 1/Nw for DMC. For the systemsconsidered here with L = 8, for |�| 1 the large deviationfunctions can be converged. However with increasing �, thestatistical effort grows significantly. For the TPS algorithm,the origin of the statistical inefficiencies can be understood bycomputing the average acceptance probability for moves madeat different times along the trajectory. Shown in Fig. 6(a) is theaverage acceptance probability for the boundary driven SEPmodel, computed for � = 0.1. We find that the acceptanceprobability is large at the ends of the trajectory and vanishingin its middle. This confinement of acceptance illustrates that

FIG. 6. Statistical efficiencies of large deviation function calculations for theSEP model. (a) The main figure shows the average acceptance ratio computedwith TPS for tN = 50 and � = 0.1. The red line is computed for forwardshifting moves and the blue line is computed for forward shooting moves.The inset is the correlation time in a number of TPS moves as a function of �.(b) The fraction of independent walkers computed with DMC for �t = 0.5 and� = 0.1. The inset is the number of correlated walks as a function of � for thesame system.

at this value of � only small perturbations to a trajectory canbe accepted. The decay of acceptance probability away fromthe ends becomes increasingly abrupt for larger �.

The efficiency of a Monte Carlo algorithm depends on howmany independent samples can be generated with a given com-putational effort. Diffusion in trajectory space will be small ifonly small changes to the trajectory are made at each MonteCarlo step. In order to quantify how correlated trajectories areduring the sequential updating of TPS, we compute a correla-tion function of current fluctuations. For trajectories separatedby ⌧ TPS moves, we define the trajectory space functionas

I(⌧) = h�Q[C0]�Q[C⌧]i, (40)

where �Q is trajectory’s current minus its ensemble average,Q hQi. This correlation function is monotonically decreas-ing, so it is uniquely characterized by its decay time which wedefine by

⌧c =

⌅ 1

0d⌧ I(⌧)/I(0) . (41)

The correlation time ⌧c is a time measured in TPS MonteCarlo moves. The inset of Fig. 6(a) shows ⌧c as a functionof �. The correlation time has been computed using movesthat result in an acceptance probability of 30% so that weare effectively probing the characteristic decay of acceptanceprobability off of the ends trajectory. Because this results inonly small changes to a trajectory to be accepted, subsequenttrajectories stay correlated over more moves for increasing �.We find that over a small range of � = 0.01–0.2, this increase isabrupt, growing over 4 orders of magnitude. To have accurateestimates, many statistically independent samples must be gen-erated by a Monte Carlo procedure, but this rapidly decreasingstatistical efficiency prohibits convergence for � > 1. As thebias being offset by the acceptance criteria is an exponentialof � times an extensive order parameter, we expect that thecorrelation time is an exponentially sensitive function of �.Because the order parameter grows with the system size ortrajectory length, the range of possible updates to the trajec-tory becomes small for large systems or tN , making studyingall but the smallest fluctuations for large systems cumbersome,if not numerically impossible.

A similar analysis can be done to understand the statisti-cal efficiencies of DMC. In the cloning algorithm, rather thanshifting the bias into acceptance criteria as TPS does, it funnelsprobability into a population dynamics. Correlations betweentrajectories result from walkers that have common ancestors.Recently, Nemoto et al. introduced a useful measure of thiscorrelation, as the fraction of independent clones, Fi(t), whichmeasures the probability that a descendant of a walker at a timet survives up to the final time tN .16 This quantity is shown inFig. 6(b) along a �t = 0.5 step for the bulk driven SEP modelwith Nw = 2000. Just as for the acceptance ratio in TPS, Fi(t)is strongly peaked at tN and decays quickly from the bound-ary. Analogously, Fi(t) reports on the number of independentwalkers and the decay time from tN sets a rate over whichwalkers become correlated.

We can quantify the statistical efficiency of DMC by com-puting the total number of correlated walkers, Nc, over the

124120-12 Ray, Chan, and Limmer J. Chem. Phys. 148, 124120 (2018)

repeated iterations of propagation and population dynamics.In terms of F i(t),

Nc = Nw(1 � Fi(0)), (42)

where F i(t) is evaluated in the limit of the time going to 0 wherewe generically find that it plateaus. This estimate is the numberof clones that share a common ancestor somewhere along thesimulation. This quantity is plotted in the inset of Fig. 6(b) asa function of �. We find that the number of correlated walkersis strongly increasing as a function of �. This means that aswe probe rarer fluctuations, fewer walkers carry significantweight. In fact, to represent the large deviation function anexponentially large number of walkers as a function of � mustbe used. As the statistical errors are inversely proportional tothe number of independent walkers,68 the estimates for large� become exponentially worse.

For both algorithms, the origin of sampling difficulties isin the exponentially decreasing overlap between the ensemblegenerated by the bare dynamics, P[C(tN )], and the ensemblethat contributes most to the large deviation function at the given�’s through the acceptance criteria in TPS or the branchingweights in DMC, P�[C(tN )]. Both algorithms propagate tra-jectories without any information from the exponential basisthat distinguishes these distributions and attempt to offset all ofthe exponential weight onto a subsequent importance samplingstep. In the case of TPS, all of the weight is put on the accep-tance criteria for making moves in trajectory space. This resultsin an acceptance rate that becomes exponentially small, andsubsequently trajectories that become correlated over expo-nentially large number of moves. In the case of DMC, all of theweight is put into the branching step. This results in a walkerpopulation that must be exponentially large in order to over-come the exponentially different walker weights. Because thetargeted ensemble is weighted by a factor that is an exponentialof a time and system size extensive quantity, these deficien-cies severely limit the current practical utility of these standardalgorithms.

C. Alternate biasing form

The final system we consider is the two dimensional,multicomponent, restricted solid-on-solid (MRSOS) model.73

This model was recently studied in the context of nonequi-librium crystal growth, where it was observed that by tuningthe growth rate the system can undergo an order-disordertransition with a concomitant critical point and regions ofbistability.74 As a two-dimensional model that exhibits criticalfluctuations, it presents an ideal testing ground for explor-ing sampling difficulties that result from large typical fluc-tuations and strong finite size effects. Also, because it hasthe potential for bistability, over short times the rate func-tion may exhibit non-convex parts, analogous to equilibriumfree energies for finite size systems at phase coexistence.Indeed few importance sampling techniques have been ableto importance sample these fluctuations efficiently. A notableexception is recent work by Klymko et al., who derived aneffective dynamics capable of sampling rare trajectories forthis model.75

The model is defined by an energy function,

E = �JX

<i,j>

sisj � µX

i

|si |, (43)

where si = {±1, 0}, J > 0 is the standard ferroelectric coupling,µ is the chemical potential for adding a spin of either sign, andhi, ji denotes a sum over only nearest neighbors. The model isevolved with a Metropolis Monte Carlo dynamics, with uni-form addition and deletion probabilities. The spin additions ordeletions are constrained to only a thin layer at the interface ofthe growing lattice. Specifically, we impose the solid-on-solidrestriction76 that does not allow for the addition or removal ofa spin with s = ±1 to a site if the difference in the new heightand the height of adjacent sites with s = ±1 exceeds 1. Thisessentially freezes the interior of the lattice, allowing for spinflips and deletions only at the interface. Starting from an initialcondition with si = 0 for all i, and a flat interface at the ori-gin, for µ > 0.5, on average the lattice will grow through thesequential additions of spins, creating a net mass current. Peri-odic boundary conditions are imposed in the direction parallelto the interface. In the direction perpendicular to the inter-face, mixed boundaries are imposed with a fixed boundaryat the origin and an open boundary in the direction of netgrowth.

The energy function in Eq. (43) is just the Ising Hamil-tonian and at equilibrium the structure should belong to atwo dimensional system independent of growth mechanism.Consequently, in a quasi-static growth limit, we expect the sys-tem to follow the physics of the two dimensional Ising modelwith a phase transition occurring at an inverse temperatureJ/kBT = ln(1 +

p2)/2 ⇠ 0.44. At J/kBT = 0.75, the system

is deep within the ordered state so that the expectation is thatthe structure should be demixed, with large domains of spinsof one sign. Remarkably the system undergoes a dynamicalphase transition dependent on the growth rate as determinedby the chemical potential, from a demixed state in the quasi-static limit to a mixed state at high growth rates.74 Exampleconfigurations on either side of the transition are shown inFig. 7. For fixed J, and small µ = 0.5, typical configura-tions are large domains of mostly spin up or down with equalprobability. At µ = 0.0, domains of spin up and spin downof many sizes span the system. For large µ = 0.5, the fastgrowth results in completely mixed lattice of spin up andspin down, with no long ranged correlations. Because of thekinetic constraint, these nonequilibrium patterns cannot easilyanneal.

Depending on which side of the dynamical transition thesystem is on, the different spatial correlations will be reflectedin the distribution of magnetization-per-particle,

M(tN ) =N(tN )X

i=1

si, (44)

where N(tN ) is the number of added spins si = ±1. At longtimes, the system relaxes to a nonequilibrium steady state withfixed growth rate N(t) / tN . In order to compute the distribu-tion function for m = M(tN )/N(tN ) in the long time limit, wecould bias ensembles of trajectories in TPS or DMC with afield conjugate to m as was done in Secs. III A and III B.However, because of the potential bistability of p(m), such alinear bias would not help sampling near m = 0 when h|m|i is

124120-13 Ray, Chan, and Limmer J. Chem. Phys. 148, 124120 (2018)

FIG. 7. Rare fluctuations of the netmagnetization for the MRSOS modelcomputed with TPS (red) and DMC(blue) for µ = (a) 0.5, (b) 0.0, and (c)0.5. The top panels show representativeconfigurations of the lattice computedwith these parameters.

finite. Instead, we choose to bias the system with a factor of theform

p,M⇤ (m) = p(m)e�(M�M⇤)2�tN (,M⇤), (45)

where the biased distribution is now a function of two param-eters, and M⇤, as is the normalization constant (, M⇤). Inthe limit that is large and positive, trajectories will be local-ized near M⇤, allowing us to sample arbitrary magnetizations,independent of the geometry of the underlying distribution.A consequence of such a bias is that (, M⇤) is no longer acumulant generating function, but inverting Eq. (45),

�(m) = �,M⇤ (m) � (M �M⇤)2/tN � (, M⇤) (46)

we still can relate the biased rate function to the unbiased one,where �(m) = ln p(m)/tN . Such a bias changes the acceptancecriteria in TPS to

Acc[Co ! Cn] = Min⇣1, e�⌦e�⌘[Cn,,M⇤]+⌘[Co,,M⇤]

⌘, (47)

where ⌦ is the path entropy that depends on the type of MCmove and

⌘[C, , M⇤] = (M[C] �M⇤)2, (48)

where M[C] is the time integrated value of the magnetizationfor the trajectory C.

Due to the quadratic bias, the branching probability forDMC has to be altered so that the long time limit adheresto the required functional form of the weight. With a linearbias, the weight is simply multiplicative, so the branchingrate can be easily computed. In the non-linear case, this isno longer possible and we must resort to the telescopic form,which requires additional book keeping. For harmonic bias,the DMC branching probability is given by

ki(t) =exp[�(�i[C0, Ct] � �i[C0, Ct��t])]PNw

j exp[�(�j[C0, Ct] � �j[C0, Ct��t])], (49)

where�i[C0, Ct] = (Mi[C0 ! Ct] �M⇤)2 (50)

is the harmonic bias up to time t. In both TPS and DMC,provided and M⇤ are chosen such that different ensem-bles of trajectories overlap, we can use histogram reweightingtogether with the relation in Eq. (46) to reconstruct �(m) deepinto the tails of the distribution.

Shown in Fig. 7 are TPS and DMC results for theMRSOS model using harmonic biases. Calculations were

accomplished for µ = 0.5, 0.0, 0.5 and J = 0.75 using systemsizes that were approximately L = 40 and tN = 200. The TPScalculations were run with = 2 and 40 windows, with M⇤/tN

chosen to cover m = 1, 1, and trajectories of 200 MC stepswere sampled with shifting and shooting moves. In all, 105

trajectories were collected for each µ. The DMC calculationswere run using Nw = 10 000 and �t = 1. For all cases a springconstant, = 2, was used and 40 values of M⇤ were spreaduniformly across the range of m = 1, 1.

For all cases studied, the DMC and TPS results are sta-tistically identical. For large µ = 0.5 as expected, the systemexhibits a single minimum, centered at m = 0, with gaussianfluctuations about the average. At intermediate driving, µ = 0,the system exhibits one broad minimum, with fluctuations thatgo like �(m) / m6 for small m. The concomitant susceptibil-ity computed under such conditions is divergent in the largesystem, long time limit, as expected for a critical transition.For small µ = 0.5, we find the system exhibits two minima,each stable basin centered at finite values of m that are equaland opposite as required by symmetry.77 If the non-convexityof �(m) is related to a surface tension in spacetime, we wouldexpect that in the limit of large tN and large L the rate func-tion becomes flat about m = 0. We indeed find this to be thecase, and further find that depending on the ratio of tN and L,interfaces can preferentially form in the smaller of the spatialor temporal directions, yielding a second metastable minimumabout m = 0, consistent with recent calculations.75

While generalizing the calculations beyond linear biasesallows TPS and DMC to sample non-convex rate functionsand study a complex model of self-assembly, the systemsizes and times are still restrictive. Indeed without additionalimportance sampling, performing finite size scaling to deter-mine the critical exponents for the transition is prohibitivelyexpensive.

IV. CONCLUSION

We have shown how large deviation functions withinnonequilibrium steady states can be computed with trajectorybased importance sampling. In particular, we have studied thesystematic and statistical errors associated with two MonteCarlo algorithms, TPS and DMC, to estimate large deviationfunctions. We found that though they are implemented in dif-ferent ways, the two algorithms sample the same ensemble oftrajectories and suffer from related difficulties.

124120-14 Ray, Chan, and Limmer J. Chem. Phys. 148, 124120 (2018)

The unifying feature of these two algorithms is that theyaugment the propagation of the bare system dynamics witha posteriori importance sampling. In the case of TPS, this isdone with a Metropolis acceptance step after proposing a newtrajectory from an existing one. In the case of DMC, the baredynamics are augmented with a population dynamics, wherebywalkers are cloned or deleted after short trajectories are gen-erated. In both cases, the proposal step of the Monte Carlochain is done without any guidance from the rare events thatcontribute to the large deviation function. As a result, for expo-nentially rarer fluctuations, the Monte Carlo chain requiresexponentially more samples of the targeted stationary distri-bution as the overlap between it and the proposed distributionbecomes exponentially small. This means that in practice lowdimensional systems, or small fluctuations, can be system-atically converged, but rare fluctuations of high dimensionalsystems are impractical to compute. We have illustrated thiswith a number of simple models where exact numerical cal-culations can be done, on and off lattice. By making simpleadjustments to the algorithm and using the standard histogramreweighting tools, we could push the algorithm to study theMRSOS model. This model exhibits a complex phase dia-gram, including a critical point. However, systematic studieson the finite size scaling properties are outside the reach ofthese current algorithms.

These results highlight a potential avenue for furtherdevelopment, where guiding functions are incorporated intothe proposal step, in analogy to what is done in quantum diffu-sion Monte Carlo.61 To do this in the context of nonequilibriumsteady states requires developing auxiliary dynamics that canpush the trajectories to the rare fluctuations that contributeto the large deviation function. In principle, for Markoviandynamics, an optimal auxiliary process always exists andcan be constructed by a canonical transformation known asthe Doob transform.48 However, to construct such a processexactly requires finding the large deviation function. In PaperII of this work, we construct approximations to the Doobtransform and use those as guiding functions for Monte Carlosampling45 and show that orders of magnitude in statistical effi-ciency can be gained by relatively simple approximate guidingfunctions.

ACKNOWLEDGMENTS

D.T.L. was supported by UC Berkeley College of Chem-istry. U.R. was supported by the Simons Collaboration onthe Many-Electron Problem and the California Institute ofTechnology. G.K.-L.C. is a Simons Investigator in Theoret-ical Physics and was supported by the California Institute ofTechnology.

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