Hard-disk dipoles and non-reversible Markov chainsPhilipp Hollmer,1, a) A. C. Maggs,2, b) and Werner Krauth3, c)1)Bethe Center for Theoretical Physics, University of Bonn, Nussallee 12, 53115 Bonn,Germany2)CNRS UMR7083, ESPCI Paris, Universite PSL, 10 rue Vauquelin, 75005 Paris,France3)Laboratoire de Physique de l’Ecole normale superieure, ENS, Universite PSL, CNRS, Sorbonne Universite,Universite de Paris, Paris, France

(Dated: 17 January 2022)

We benchmark event-chain Monte Carlo (ECMC) algorithms for tethered hard-disk dipoles in two dimensionsin view of application of ECMC to water models in molecular simulation. We characterize the rotationdynamics of dipoles through the integrated autocorrelation times of the polarization. The non-reversiblestraight, reflective, forward, and Newtonian ECMC algorithms are all event-driven, and they differ only intheir update rules at event times. They realize considerable speedups with respect to the local reversibleMetropolis algorithm. We also find significant speed differences among the ECMC variants. NewtonianECMC appears particularly well-suited for overcoming the dynamical arrest that has plagued straight ECMCfor three-dimensional dipolar models with Coulomb interactions.

I. INTRODUCTION

Markov-chain Monte Carlo1 (MCMC) is a computa-tional method for sampling high-dimensional probabilitydistributions π including the Boltzmann distribution ofstatistical physics. In MCMC, an initial sample x, at atime t = 0, is drawn from a distribution π0 rather thanfrom the “equilibrium” distribution π. The sample x, attime t, moves to the sample y, at time t+ 1, with a con-ditional probability Pxy, element of a time-independent

transition matrix P . The probability distribution πt

then moves to πt+1 = πtP . The intermediate proba-bility distributions are usually not known explicitly, butthey evolve towards π for t→∞. The transition matrixP must satisfy a balance condition that expresses the sta-tionarity of π. Reversible transition matrices satisfy thedetailed-balance condition πxPxy = πyPyx for all x and y.Detailed balance is equivalent to the statement that theequilibrium flow Fxy = πxPxy from x to y equals the re-verse flow from y to x. Non-reversible MCMC algorithmssatisfy a weaker global-balance condition πx =

∑y πyPyx

for all x. Under conditions of irreducibility and aperiod-icity,2 they converge to the equilibrium distribution πwith non-zero net flows Fxy − Fyx. Samples are thendrawn from the equilibrium distribution π for t → ∞by a non-equilibrium random process with non-vanishingnet flows.

In past decades, research and applications have focusedalmost exclusively on reversible Markov chains (and someclose relatives, such as sequential schemes3,4). ReversibleMarkov chains are straightforward to set up for arbitraryprobability distributions π, and are easy to conceptual-ize, in particular because of the real-valued eigenvalue

a)Electronic mail: hoellmer@physik.uni-bonn.deb)Electronic mail: anthony.maggs@espci.frc)Electronic mail: werner.krauth@ens.fr

spectrum of their transition matrices. Popular reversibleMarkov chains include the Metropolis-1 or heat-bath5–7

algorithms. Interest in non-reversible Markov chains hasincreased in recent years, ever since it was understood8,9

that such algorithms often approach equilibrium fasterthan reversible Markov chains. Systematic non-reversibleMarkov-chain schemes are becoming available.10,11

Event-chain Monte Carlo11–13 (ECMC) is a family ofnon-reversible local Markov chains that have been ap-plied to, e.g., particle14 and spin systems,15–17 poly-mers18,19 and field-theoretical models.20 ECMC algo-rithms are defined in continuous time, and they can be in-terpreted as piecewise deterministic Markov processes.21

They have found applications in Bayesian statistics.22,23

A number of features set ECMC algorithms apart fromreversible Markov chains. First, they may reach the“equilibrium” distribution π on fast ballistic time scalesrather than on the diffusive time scales that are typ-ically associated with local reversibility. Considerablespeedups were demonstrated in analytically solvable testcases8,9,24 and recovered in applications of practical rele-vance.13,14,19,20 Second, ECMC, for a given model (prob-ability distribution π), covers closely related local MCMCalgorithms that can behave quite differently. Algorithmsadopt a wide range of factor sets for the breakup of theBoltzmann distribution. On a finer scale, even for a givenset of factors, different variants may have widely differ-ent behaviors, as we will discuss in the present paper.Third, ECMC need not evaluate the ratio πy/πx to movefrom a sample x to a sample y (in other words, need notevaluate the change in energy), because the factors arestatistically independent.12,25

The sampling of the Boltzmann distribution with-out evaluating the energy is implemented in a general-purpose open-source ECMC application26 for classicalmolecular simulation in Coulomb systems, where theevaluation of the energy represents for other methodsthe computational bottleneck. In the test case of three-dimensional water with the simple-point-charge-flexible-

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water (SPC/Fw) potential27 and the “periodic” variantof straight ECMC (see Section III B for a definition),the simulated water molecules however resisted rotationand the polarization remained dynamically arrested forlong times.28 The present paper studies many variantsof ECMC for a simpler model of polar molecules, andsuggests that its periodic variant is ill-suited to thesesystems.

In a generic Markov chain, the sample space Ω is dis-tinct from the moves ∆. The latter are then part of amove set and lead from samples x ∈ Ω to samples y ∈ Ω.In the local reversible Metropolis algorithm for particlesystems, for example, ∆ = (δ, i) consists in a small ran-dom displacement δ of a random particle i. ECMC, incontrast, belongs to the class of lifted Markov chains,8,9

in which parts of the move set are integrated into a“lifted” sample space. In the above example, the pro-posed move ∆′ = (δ′, i′) at time t + 1, after a move ∆,at time t, is then no longer independently sampled, butit rather depends on the lifted configuration, and in par-ticular on ∆. The probability distribution lives itself inan extended space. Nevertheless, in the cases that weconsider here, the stationary probability distribution inthe lifted space factorizes into the original distributionπ and a distribution for the lifting variables, while thetransition matrix non-trivially couples the two sectors.

In this paper we caricature the SPC/Fw water modelas tethered hard-disk dipoles. This replaces a three-dimensional model by a two-dimensional one, and lumpsthe vibrational, bending, Lennard-Jones, and Coulombinteractions into a hard-disk potential. The simplifieddipole model retains the polar nature of molecules, and itcan be simulated orders of magnitude faster than the sys-tem of flexible Coulomb dipoles. It also greatly simplifiesthe implementation of the different algorithms. This al-lows us to scan straight,11,29 reflective,11 forward,30 andNewtonian31 ECMC with thousands of parameter sets.We benchmark the ECMC algorithms against the localMetropolis algorithms with different displacement sets.We characterize the speed of algorithms via the autocor-relation of the polarization, the mode that with ECMCequilibrates slowly in the SPC/Fw water model.28

For dipole-model parameters that roughly correspondto those of SPC/Fw water, we find a 50-fold speedup forthe fastest non-reversible ECMC algorithm with respectto the optimized local reversible Metropolis algorithm.We also find a more than order-of-magnitude speed dif-ference among the different ECMC variants. The dif-ferent versions of straight ECMC are clearly the slowestones at low density, while they are comparable with for-ward and reflective ECMC at high densities. NewtonianECMC is by far the fastest. We suggest that this is dueto its absence of intrinsic parameters.

The content of this paper is as follows: In Section II,we define the two-dimensional tethered hard-disk dipolemodel and motivate its parameters with respect to phys-ical systems of the SPC/Fw model in three dimensions.We also discuss the polarization, which tracks the abil-

ity of dipoles to rotate. In Section III, we discuss thereversible and non-reversible MCMC algorithms that wehave implemented and illustrate their behavior for thecase of a single dipole. We also discuss subtle aspects con-cerning the irreducibility of the Markov chains. In Sec-tion IV, we benchmark the different variants of ECMCagainst the local reversible Metropolis algorithm for arange of densities and system sizes, and we evidence thesuperiority of the non-reversible ECMC algorithms. InSection V, we project how our conclusions can be ex-tended to the more complex water models.

II. HARD-DISK DIPOLE MODEL

We consider tethered dipoles in a two-dimensionalsquare box of sides L with periodic boundary conditions.Each dipole consists of two hard disks of radius σ, witha flat inner-dipole interaction that constrains its exten-sion ρ (the separation of the disk centers) between thecontact distance r = 2σ and the tether length R (seeFig. 1a). Dipole configurations without overlapping disksand with all dipole extensions between r and R all havethe same Boltzmann weight, whereas all other configura-tions have zero statistical weight. As the minimum dipoleextension at contact equals 2σ (see Fig. 1a), each dipoleconfiguration is also a configuration of hard disks. Anydipole system is parameterized by the number of dipolesN , the tether ratio η = R/r, and the hard-disk densityD = 2Nπσ2/L2.

In a single three-dimensional water molecule at roomtemperature (as for example described in the SPC/Fwmodel), the oxygen–hydrogen distance fluctuates by ∼2.3 % and the hydrogen–hydrogen distance by ∼ 3.4 %(see Fig. 1b). In the hard-disk dipole model, we thusadopt a tether ratio η = 1.1, for which the ∼ 2.75 %fluctuations of the extension are quite similar. The tetherlengthR is thus only 10% larger than the contact distancebetween two disks, so that two dipoles cannot lock intoa crossed state that would be difficult to disentangle intwo dimensions. Our results for the dynamics of two-dimensional models may well extend to three dimensions.

We measure dynamical information through the inte-grated autocorrelation functions of components of the to-tal polarization p:

p =

N∑i=1

pi, (1)

where pi is the oriented separation vector between thetwo disks in the ith dipole, possibly corrected for periodicboundary conditions (see Fig. 1c and d).

For N = 1, analytic results are available for the dy-namics of some of the algorithms discussed here.29 Thecenter-of-mass motion of the single dipole then factorsout, and the sample space Ω simply corresponds to thepossible values of the polarization vector, in other wordsto the two-dimensional ring with inner radius r and outer

3

FIG. 1. Tethered hard-disk dipole model. (a): Dipoles withtwo hard disks of radius σ, and minimum extension r = 2σand tether length R. (b): Probability distribution of theoxygen–hydrogen (OH) and hydrogen–hydrogen (HH) bondlengths for a single SPC/Fw water molecule at room tem-perature. (c): Five dipoles with polarization pi. (d): Totalpolarization p of (c), together with its probability distribu-tion.

radius R (see Ref. 29). The single-dipole polarization pis uniformly distributed in this ring.

III. REVERSIBLE AND NON-REVERSIBLE MCMC

In the present section, we introduce MCMC algorithmsthat will be used in Section IV. We also illustrate themfor a single dipole (N = 1) and review some of their fun-damental properties. We only consider algorithms whereat each moment a single disk moves, as this can alsobe realized within ECMC in the presence of long-rangeCoulomb interactions.25,26 Dipole rotations—that relaxthe polarization—are thus pieced together from subse-quent displacements of single disks. This has alreadyproven efficient in ECMC for polymer models and forelongated hard needles.18,19 Explicit dipole rotations ap-pear difficult to put into place in this context.

The local MCMC moves discussed here feature smallsingle-disk displacements at each time step (for theMetropolis algorithm) or continuous moves in time (forthe ECMC algorithms). Again, this is motivated by thefuture applications to more complicated systems such aswater, for which non-local moves including cluster up-dates32,33 appear out of reach. Although we thus restrictour attention to a subclass of MCMC algorithms with

very similar design principles, we will show in Section IVthat their properties differ substantially.

In the ECMC algorithms, a single active disk moveswith a constant velocity in a given direction. Thestraight-line trajectories stop at events which correspondto hard-disk collisions or to the moment when the dipoleextension reaches the tether length. In addition, resam-plings take place at predefined MCMC times, which areseparated by a chain time τchain. Resamplings also inter-rupt the straight-line trajectory, and for example draw anew velocity and a new active disk.

A. Reversible local Metropolis algorithms

The local Metropolis algorithm,1 for the dipole model,selects a random disk i at each time step. For the cross-shaped displacement set, the proposed move for i is sam-pled uniformly between −δ and δ with range δ, randomlyin x- or in y-direction. For the square-shaped displace-ment set, the proposed move is uniformly sampled froma square of sides 2δ centered at zero, with both the xand the y components of the displacement sampled in-dependently between −δ and δ. In both cases, the moveis accepted if it violates no constraints. If the move isrejected, the configurations at times t and t + 1 are thesame. The Metropolis algorithm, with the given displace-ment sets, satisfies the detailed-balance condition. ForN = 1, the polarization p performs a random walk inthe sample space Ω. The algorithm can be considered ir-reducible for all N , though some blocked configurations34

may pose problems.35

B. Straight ECMC

In straight ECMC,11 the velocity is of constant abso-lute value, and it changes only at resamplings. We an-alyze several variants for updating the angle φ that de-scribes the velocity in two dimensions. In the “periodic”variant, the angle alternates between φ = 0 and φ = π/2.In arbitrary dimension, the velocity cycles through theunit vectors aligned with the positive coordinate axes.The periodic variant, used in most previous works, is im-plemented in the general-purpose ECMC application.26

In the present paper, we also analyze the “random” vari-ant where φ is uniformly sampled in the interval [0, 2π)and, finally, the “sequential” variant,29 where φ is in-cremented by a constant ∆φ at each resampling. Thepolarization trajectory between resamplings of straightECMC is a straight line. In the periodic variant, subse-quent straight-line trajectories are at right angles. Theyare at an angle ∆φ in the sequential variant (see Fig. 2afor an example in N = 1). The described variants havevery different time evolutions. The polarization trajec-tories of the sequential variant, for example, persistentlyrotate for N = 1, with a total rotation angle that divergesas ∆φ goes to zero.29

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FIG. 2. Sample trajectories for a single tethered dipole. Thepolarization p is the oriented vector between the two diskcenters. (a): Sequential variant of straight ECMC.29 (b):Reflective ECMC.11 (c): Forward ECMC.30 (d): NewtonianECMC.31

At a resampling of straight ECMC at time tres (usu-ally a multiple of τchain), the angle φ is updated, butthe active disk must also be chosen randomly in orderto satisfy global balance. To motivate this, we show forN = 1 that the two disks 1 and 2 must be active withthe same probability at time tres. Up until the next re-sampling, all polarizations p(t) lie on a segment of thering Ω, and each of the two directions of motion on thatsegment corresponds to one of the two disks being ac-tive (see Fig. 3a). Global balance requires the flows intop(t) to be independent of its position. It can be reachedfrom two polarizations p(tres) and p′(tres). Depending onthe value of p(t), the corresponding polarizations p(tres)and p′(tres) either both correspond to disk 1 being ac-tive at tres, or both to disk 2, or one to 1 and one to2 (see Fig. 3a). Disks 1 and 2 must thus be equallylikely (p1 = p2) to be active right after the resampling, a“random”-mode condition that is precisely implementedby the resampling of the active disk at tres.

To illustrate the relevance of the random-mode require-ment, we test for N = 1 the case when the active diskis maintained at a resampling while the angle φ is in-cremented in the sequential variant of straight ECMC(“snake” mode). Clearly, snake mode is incorrect, asit samples Ω non-uniformly (p1 6= p2, see Fig. 3b). Itseems to be incorrect even when both disks happen tobe initially active with practically the same probability(p1 = p2, see Fig. 3c). In both cases, the snake-modetrajectory is repetitive, and lack of irreducibility in thelifted sample space Ω × φ induces a non-uniform dis-tribution in Ω. For N > 1 at very low density, sequential

FIG. 3. Global balance and irreducibility for sequentialECMC. (a): Flow into polarization p(t) from p(tres) andp′(tres) on a segment of Ω as in Fig. 2a. Events take place atthe boundaries of the segment, and the active disk changes.(b): Single-dipole distribution π(p/σ) for sequential ECMCwith small ∆φ. The snake-mode distribution is incorrect, andthe probability p1 that disk 1 is active right after a resamplingis smaller than p2. (c): Same as (b) but for different initialconfiguration and chain time. The snake-mode distribution isincorrect although both disks are equally likely to be activeafter a resampling (p1 = p2).

ECMC in snake mode might not be irreducible, becausea single dipole deterministically rotates on a closed tra-jectory. However, no difference between snake mode andrandom mode is detected for N > 1 at the higher densi-ties considered in Section IV.

C. Reflective ECMC

Reflective ECMC11 differs from the straight ECMConly in its handling of events. At an event, the velocityof the incoming active disk is reflected by the line con-necting the active and the target disks, and it becomesthe velocity of the target disk. The absolute value of thevelocity is thus preserved. For N = 1, the polarization

5

follows straight lines which are symmetrically reflectedoff the inner and outer ring boundaries, with a singlereflection angle α (see Fig. 2b).

Reflective ECMC satisfies global balance for any num-ber of dipoles. For N = 1, and for an initial velocitythat is nearly perpendicular to the separation betweenthe two disks, the polarization may follow a “whispering-gallery-mode” trajectory which never visits small separa-tions ρ & r (compare with Ref. 23, Fig. 3), showing thatresamplings of the active disk and of its velocity angleare required for irreducibility. For all initial velocitiesat N = 1, even for those that visit all of sample spaceΩ, the probability distribution in the lifted sample spacedoes not separate into independent distributions, and thesampled stationary distribution in Ω is non-uniform, andthus incorrect. With periodic resamplings, the station-ary distribution is uniform in Ω. For N > 1 dipoles,in a large enough box, reflective ECMC fails to be ir-reducible without resamplings, as the dynamics is thendeterministic and a single dipole may effectively rotatein a stationary closed trajectory. Nevertheless, at highenough density, we have not observed any difference instatic observables with or without resamplings. It thusappears safe to omit them in practical simulations. Aswe will see in Section IV, resamplings do not improve thedecorrelation. Without resamplings, reflective ECMC isfree of intrinsic parameters, and somewhat easier to setup. Computational overhead is avoided.

In previous work on hard disks11 (rather than hard-disk dipoles), reflective ECMC was benchmarked againststraight ECMC and found to be considerably slower.In contrast, in the dipole model studied here, reflectiveECMC will prove more powerful than straight ECMC,even when the latter runs with optimized intrinsic pa-rameters.

D. Forward ECMC

Forward ECMC30 resembles reflective ECMC in that,after an event, the target-disk velocities of both algo-rithms are located in the same quadrant of the coordi-nate system with axes parallel and orthogonal to theline connecting the disks at the event. All velocitiesare of unit absolute value, but forward ECMC incor-porates a random element into each event. More pre-cisely, the component orthogonal to the line connectingthe disks at contact is uniformly sampled between 0 and1 (while reflecting the orthogonal orientation). Its par-allel component is determined so that the velocity is ofunit norm. For a single dipole, the polarization describesstraight lines that are reflected off the inner and outerring boundaries. However, the outgoing reflection angleis non-deterministic (see Fig. 2c). Forward ECMC is ir-reducible even for N = 1, and it requires no resamplings.Also, as we will show in Section IV, resamplings of thevelocity and the active disk in intervals of the chain timedo not speed up the algorithm.

E. Newtonian ECMC

Newtonian ECMC31 mimics event-driven moleculardynamics.36 All disks have velocity labels in addition totheir positions. The label of the active disk indicates thetime derivative of its position, that is, its displacementin time. All other disks are stationary. Events (includingthose where the maximum dipole separation is reached)take place as in molecular dynamics, with all velocity la-bels treated on an equal footing, and with equal massesfor all disks. The identities of the active and the targetdisks are interchanged in the event. During the simula-tion, the absolute value of the active-disk velocity varies.The equilibrium distribution of all velocity labels is uni-form on the 2N -dimensional unit sphere, and the root-mean-square velocity vrms is conserved. The event rateper unit distance of the Newtonian ECMC equals the oneof the other variants, but the event rate per unit timeis smaller by

√π/2, because of the difference between

vrms and the mean absolute value for the two-dimensionalGaussian distribution of velocities. All the velocity la-bels are used as lifting variables. At a resampling, theselabels must be sampled from the exact equilibrium distri-bution (the rescaled Maxwell–Boltzmann distribution forall disks). Without resampling, the choice of the initialvelocity labels is arbitrary.

For N = 1, the polarization trajectory of Newto-nian ECMC reflects off the inner and outer ring bound-aries but considers both velocity labels at each event(see Fig. 2d). Without resamplings, Newtonian ECMCbreaks irreducibility for this single dipole, and it sam-ples an incorrect stationary probability distribution. ForN ≥ 2, we find no influence of observable means onthe resampling rate, so that the irreducibility problemis probably again due to the high symmetry of the ringgeometry in Section 2. In the larger-N dipole systemsin Section IV, we furthermore notice that the algorithmbecomes faster in the limit of infinite resampling time.

IV. AUTOCORRELATIONS FOR N DIPOLES

In the present section, we characterize the local MCMCalgorithms of Section III via the autocorrelation dynam-ics of the polarization. All correlation times τint refer toaveraged integrated autocorrelation times of single com-ponents of the polarization. The unit of time correspondsto a single trial move for the Metropolis algorithm or tothe mean event time for ECMC. In ECMC, the compu-tational effort is O(1) per event, as it is per move inthe Metropolis algorithm. For this reason, no effort ismade to compare actual CPU times, which would dependon the implementation. Statistical errors are estimatedfrom the difference between autocorrelation times of thex and y components of the polarization which, by sym-metry, must be the same. For ECMC, we also take intoaccount the uncertainty in the mean event time. SingleMCMC-run times, for each parameter set, are at least

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0.0 0.4 0.8 1.2range /

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(b)FIG. 4. Correlation time τint of the Metropolis algorithmwith cross- and square-shaped displacement sets for N = 81dipoles at density D = 0.70. (a): τint vs. the range δ. (b): τintvs. acceptance rate. The minimum of τint provides the refer-ence in Fig. 5, that the ECMC algorithms are benchmarkedagainst.

three orders of magnitude longer than τint.

A. Intrinsic parameters of Metropolis and straight ECMC

The correlation time of the local Metropolis algorithmdepends on its intrinsic parameters, namely the range δand the displacement set (see Fig. 4a). The acceptancerate decreases with increasing δ. The correlation timecan thus be expressed as a function of the acceptancerate. As in many comparable models, the “one-half”rule37 is roughly respected, and the Metropolis algorithmconverges best for a rejection rate on the order of 50%.For N = 81 dipoles at density D = 0.70, we observea broad optimum between 20 % to 40 % for the square-shaped displacement set and an optimum close to 50%for the cross-shaped displacement set (see Fig. 4b).

All ECMC variants allow for resamplings so thattheir correlation times depend on the intrinsic param-eter τchain. The performance of the sequential variant ofstraight ECMC also depends on the angle increment ∆φ.Without resamplings, straight ECMC is not irreducibleand its correlation time is infinite. The optimum τint isthus at a finite τchain, to be obtained by fine-tuning (seeupper curves in Fig. 5). In its sequential variant, thecases ∆φ = π/2 and ∆φ = π/3 make the velocity cyclethrough a few values only. For these cases, we notice thatτint has two local minima, both of which are rather large.In the absence of a heuristic for the choice of intrinsicparameters, straight ECMC requires explicit fine-tuning,which increases its computational complexity.

As discussed in Section III, reflective, forward andNewtonian ECMC appear irreducible for N > 1 evenwithout resampling and are fastest in this limit τchain →∞ (see lower curves in Fig. 5). This was also ob-served, e.g., for reflective ECMC in two-dimensionalhard-disk systems,11 and for Newtonian ECMC in three-dimensional hard spheres.31 For moderate values of

FIG. 5. Correlation times of ECMC variants for differentintrinsic parameters (chain time τchain, angle increment ∆φ)for N = 81 dipoles at density D = 0.70. Optimal values of τintare highlighted on the right y-axis. For reflective, forward,and Newtonian ECMC, they agree with those for τchain →∞. The optimal performance of the Metropolis algorithm isindicated as the benchmark reference (see Fig. 4). The insetillustrates the evolution of τint for large τchain for NewtonianECMC.

τchain, Newtonian ECMC is comparable to other vari-ants. However, for much larger τchain, its correlationtimes again decrease strongly (see inset of Fig. 5). ForN = 81 dipoles at density D = 0.70, the variants thatrequire no fine-tuning perform best. Newtonian ECMCwithout resampling accelerates with respect to the localMetropolis algorithm by a surprising factor 50 at den-sity 0.70, that seems to further increase at even higherdensity (see Section IV B).

70.0 0.4 0.8 1.2range /

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FIG. 6. Performance of the Metropolis algorithm for N = 81dipoles (optimized intrinsic parameters). (a): Correlationtime τint vs. density D for the cross- and square-shaped dis-placement sets. (b): Speedup for the square-shaped displace-ment set compared to the cross-shaped displacement set.

B. Density and size dependence of speedups

For both versions of the Metropolis algorithms, τintnaturally increases at higher densities and climbs steeplyfor D & 0.70, around the liquid–hexatic phase-transitiondensity of the hard-disk system without the tether con-straint (see Fig. 6a, again for N = 81). For theMetropolis algorithm, the square-shaped displacementset is somewhat preferable over the cross-shaped displace-ment set for small densities, but the two become equallyfast at large densities (see Fig. 6b). This illustratesthat, while step-size control is a key feature in reversibleMCMC, the details of a local reversible algorithm do notreally depend on the displacement sets. For simplicity,we chose the optimal acceptance rates for D = 0.70 forthese runs at other densities (see Fig. 4).

For the correlation times of the straight ECMC vari-ants, we perform systematic scans as in Section IV A,for each density at N = 81. For forward, reflective, andNewtonian ECMC, we use τchain = ∞ without any fine-tuning. For all ECMC variants, τint increases with den-sity (see Fig. 7a). The periodic variant, which is fastestfor two-dimensional hard disks, is the slowest variant forhard-disk dipoles. The random and sequential variantsare somewhat faster. Reflective and forward ECMC re-semble each other in performance at all D. While theyare considerably faster than the straight variants at lowD, this gap vanishes at D = 0.72. Newtonian ECMC isthe fastest algorithm for all densities and, as mentioned,it requires no fine-tuning. All ECMC algorithms outper-form the Metropolis algorithm by a considerable mar-gin (see Fig. 7b). With increasing density, the speedupof reflective and of forward ECMC drops from roughly30 to 15, while the speedup of the straight ECMC vari-ants increases with density until they become compara-ble. These trends were similarly observed near the liquid–hexatic transition of two-dimensional hard disks for peri-odic and reflective ECMC11 (where the periodic variant,however, was considerably faster than reflective ECMC

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FIG. 7. Performance of ECMC variants and of the Metropo-lis algorithm with a square-shaped displacement set (whereapplicable: with optimized intrinsic parameters). (a): Cor-relation time τint for N = 81 dipoles vs. density D. (b):Speedup of ECMC with respect to the Metropolis algorithm.(c): Correlation time τint at density D = 0.70 vs. number ofdipoles N .

at high densities). The speedup realized by NewtonianECMC improves slightly from low to high densities, aswas likewise found for three-dimensional hard spheres.31

We observe the highest speedup of roughly a factor of 60at the highest density D = 0.72 that we studied.

We finally study the dependence of the correlation timeτint of the number of dipoles N at the density D = 0.70.For the reflective, forward and Newtonian ECMC vari-ants, we simply set τchain = ∞, given our findings atN = 81. For periodic, random and sequential ECMC,we infer the optimal τchain and ∆φ from the N = 81 case(performing occasional sweeps through intrinsic param-eters as cross-checks). For the Metropolis algorithm, weused the optimal acceptance rate of N = 81. We observethat τint increases linearly with N for all algorithms (see

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Fig. 7c). This implies that the observed speedups of thenon-reversible algorithms stay constant.

V. CONCLUSIONS AND OUTLOOK

In this paper, we have systematically studied localMCMC algorithms for complex molecules that we car-icatured as hard-disk dipoles with parameters inspiredby the SPC/Fw water system. Moving from three to twospatial dimensions and from a liquid of charged moleculesto a model of hard-disk dipoles has greatly simplified thealgorithm development and its implementation. This al-lowed us to scan thousands of parameters sets for differ-ent ECMC variants and to benchmark them against thereversible local Metropolis algorithm. At this exploratorystage, this would have required prohibitive computing re-sources for the original three-dimensional model.28 Thebroad spread of the performance of different ECMC ver-sions came as a surprise. We found that algorithms withintrinsic parameters perform considerably less well thanothers and demonstrated a 60-fold speedup of NewtonianECMC with respect to the Metropolis algorithm. Ourstudy has also brought out subtleties of ECMC, that arehidden in simple liquids. For the case of a single dipole,we discovered strict requirements for resampling in orderto ensure irreducibility of straight, reflective and Newto-nian ECMC. However, these requirements did not seemto play a role for denser systems of dipoles. Event-basedrandomness as in forward ECMC (or as in all ECMC sim-ulations of soft interactions) strictly ensures irreducibil-ity.

We believe that our observations will carry over fromthe caricature two-dimensional dipoles to the three-dimensional SPC/Fw systems and related models. Thereare two reasons why we have not yet studied this systemdirectly. One is, as mentioned, the scale of the com-puting requirements for a full-fledged three-dimensionalscan. The other is that further algorithm development isstill required. The cell-veto algorithm38 (which reducesthe computational complexity of the Coulomb problem toO(N logN) per sweep of events, without evaluating theCoulomb energy) is in the present version of our open-source project only implemented for the periodic variantof straight ECMC. However, we expect it to generalize toall ECMC variants for flexible water molecules with anexplicit Coulomb interaction. The benchmark againstMetropolis MCMC will then be much more favorable, asthe change of the Coulomb energy can there only be com-puted inO(N3/2) per sweep of moves.39 The key questionwill be whether the dynamic arrest of straight ECMCfor the three-dimensional water system can be overcomeas in the two-dimensional hard-disk dipole model stud-ied here. The extreme dependence of the performanceof non-reversible MCMC on details of the algorithm wasalso evidenced in the escape times from a tightly con-fined initial configuration,35 which might be relevant forovercoming dynamic arrest.

Finally, we point out that any tethered-dipole config-uration is also a valid hard-disk configuration, with anadded upper limit on the distance between pairs of harddisks. For the tethered dipole system, the global hexaticorientational order parameter ψ6 (see Refs. 14 and 40),at densities D & 0.70, decorrelates faster than for thehard-disk system which there already enters the mixedhexatic–fluid phase. This observed effect might allowfor an improved sampling algorithm for hard disks. Itis also unclear why straight ECMC performs worse fordipoles than the reflective, forward, and Newtonian vari-ants, while for simple hard disks straight ECMC is clearlythe most efficient.

ACKNOWLEDGMENTS

P. H. acknowledges support from the Studienstiftungdes deutschen Volkes and from Institut Philippe Meyer.W. K. acknowledges support from the Alexander vonHumboldt Foundation.

ADDITIONAL REMARKS

The figures in this paper were prepared with the Mat-plotlib41 and Visual Molecular Dynamics42 (VMD) soft-ware packages. Integrated autocorrelation times werecomputed by the Python emcee package.43

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