Generalized event-chain Monte Carlo: Constructing rejection-free global-balance algorithms from infinitesimal steps

Necessary and sufficient symmetries in Event-Chain Monte Carlo with generalized flows and application to hard dimers

Event-Chain Monte Carlo (ECMC) methods generate continuous-time and non-reversible Markov processes, which often display significant accelerations compared to their reversible counterparts. However, their generalization to any system may appear less straightforward. In this work, our aim is to distinctly define the essential symmetries that such ECMC algorithms must adhere to, differentiating between necessary and sufficient conditions. This exploration intends to delineate the balance between requirements that could be overly limiting in broad applications and those that are fundamentally essential. To do so, we build on the recent analytical description of such methods as generating piecewise deterministic Markov processes. Therefore, starting with translational flows, we establish the necessary rotational invariance of the probability flows, along with determining the minimum event rate. This rate is identified with the corresponding infinitesimal Metropolis rejection rate. Obeying such conditions ensures the correct invariance for any ECMC scheme. Subsequently, we extend these findings to encompass schemes involving deterministic flows that are more general than mere translational ones. Specifically, we define two classes of interest of general flows: the ideal and uniform-ideal ones. They, respectively, suppress or reduce the event rates. From there, we implement a comprehensive non-reversible sampling of a system of hard dimers by introducing rotational flows, which are uniform-ideal. This implementation results in a speed-up of up to ∼3 compared to the state-of-the-art ECMC/Metropolis hybrid scheme.

Sampling Efficiency of Transverse Forces in Dense Liquids

Sampling the Boltzmann distribution using forces that violate detailed balance can be faster than with the equilibrium evolution, but the acceleration depends on the nature of the nonequilibrium drive and the physical situation. Here, we study the efficiency of forces transverse to energy gradients in dense liquids through a combination of techniques: Brownian dynamics simulations, exact infinite-dimensional calculation, and a mode-coupling approximation. We find that the sampling speedup varies nonmonotonically with temperature, and decreases as the system becomes more glassy. We characterize the interplay between the distance to equilibrium and the efficiency of transverse forces by means of odd transport coefficients.

Asymmetry in three-site relaxation exchange NMR

The asymmetry of peak integrals in 2D relaxation maps of exchange between three sites indicates circular flow between the relaxation sites. This disagrees with the detailed balance according to which the exchange between any pair of sites must be balanced in terms of thermodynamic equilibrium. Confined diffusion of particles jumping randomly on a 2D checkerboard grid to any of their eight neighbor positions and confined gas diffusion were modeled in Monte Carlo simulations to explore the impact of topological constraints on particle exchange between three pools. Both models produce density variations across the pore and reveal that up to 1 % of the molecules move in circular paths between the relaxation pools. This motion is driven by different features of either algorithm. It is silent in terms of thermodynamic equilibrium, confirming that multi-site exchange maps are symmetric in this case. The coherent flux is argued to result from stochastic pore resonance related to diffusion eigenmodes. If it can be driven experimentally by external time-varying electric, magnetic, or ultrasonic fields, this may be a way to enhance heterogeneous catalysis.

Hard-disk pressure computations-a historic perspective

We discuss pressure computations for the hard-disk model performed since 1953 and compare them to the results that we obtain with a powerful event-chain Monte Carlo and a massively parallel Metropolis algorithm. Like other simple models in the sciences, such as the Drosophila model of biology, the hard-disk model has needed monumental efforts to be understood. In particular, we argue that the difficulty of estimating the pressure has not been fully realized in the decades-long controversy over the hard-disk phase-transition scenario. We present the physics of the hard-disk model, the definition of the pressure and its unbiased estimators, several of which are new. We further treat different sampling algorithms and crucial criteria for bounding mixing times in the absence of analytical predictions. Our definite results for the pressure, for up to one million disks, may serve as benchmarks for future sampling algorithms. A synopsis of hard-disk pressure data as well as different versions of the sampling algorithms and pressure estimators are made available in an open-source repository.

Lifted directed-worm algorithm

Nonreversible Markov chains can outperform reversible chains in the Markov chain Monte Carlo method. Lifting is a versatile approach to introducing net stochastic flow in state space and constructing a nonreversible Markov chain. We present here an application of the lifting technique to the directed-worm algorithm. The transition probability of the worm update is optimized using the geometric allocation approach; the worm backscattering probability is minimized, and the stochastic flow breaking the detailed balance is maximized. We demonstrate the performance improvement over the previous worm and cluster algorithms for the four-dimensional hypercubic lattice Ising model. The sampling efficiency of the present algorithm is approximately 80, 5, and 1.7 times as high as those of the standard worm algorithm, the Wolff cluster algorithm, and the previous lifted worm algorithm, respectively. We estimate the dynamic critical exponent of the hypercubic lattice Ising model to be z≈0 in the worm and the Wolff cluster updates. The lifted version of the directed-worm algorithm can be applied to a variety of quantum systems as well as classical systems.

Large-scale dynamics of event-chain Monte Carlo

Event-chain Monte Carlo (ECMC) accelerates the sampling of hard-sphere systems, and has been generalized to the potentials used in classical molecular simulations. Rather than imposing detailed balance on the transition probabilities, the method enforces a weaker global-balance condition in order to guarantee convergence to equilibrium. In this paper, we generalize the factor-field variant of ECMC to higher space dimensions. In the two-dimensional fluid phase, factor-field ECMC saturates the lower bound z=0 for the dynamical scaling exponent for local dynamics, whereas molecular dynamics is characterized by z=1 and local Metropolis Monte Carlo by z=2. In the presence of hexatic order, factor fields are not found to speed up the convergence. We note that generalizations of factor fields could couple to orientational order.

Reverse order-disorder transition of Janus particles confined in two dimensions

Janus particles with different patch sizes, confined to two dimensions, generate a series of patterns of interest to the field of nanoscience. Here we observe reverse melting, where for some densities the system melts under cooling. For a broad range of hydrophobic patch sizes (60^{∘}<θ_{0}<90^{∘}), a reentrant transition from solid to liquid and then to an ordered phase emerges as temperature (T) decreases due to the formation of rhombus chains at low T. This reentrant phase has pseudo long-range orientational order but short-range translational order, similar to a hexatic phase. Our work provides guidelines to study the melting and assembly of Janus particles in two dimensions, as well as mechanisms to generate phases with specific symmetry.

Newtonian Event-Chain Monte Carlo and Collision Prediction with Polyhedral Particles

Polyhedral nanocrystals are building blocks for nanostructured materials that find applications in catalysis and plasmonics. Synthesis efforts and self-assembly experiments have been assisted by computer simulations that predict phase equilibria. Most current simulations employ Monte Carlo methods, which generate stochastic dynamics. Collective and correlated configuration updates are alternatives that promise higher computational efficiency and generate trajectories with realistic dynamics. One such alternative involves event-chain updates and has recently been proposed for spherical particles. In this contribution, we develop and apply event-chain Monte Carlo for hard convex polyhedra. Our simulation makes use of an improved computational geometry algorithm XenoSweep, which predicts sweep collision in a particularly simple way. We implement Newtonian event chains in the open-source general-purpose particle simulation toolkit HOOMD-blue for serial and parallel simulation. The speedup over state-of-the-art Monte Carlo is between a factor of 10 for nearly spherical polyhedra and a factor of 2 for highly aspherical polyhedra. Finally, we validate the Newtonian event-chain algorithm by applying it to a current research problem, the multistep nucleation of two classes of hard polyhedra.

From predictive modelling to machine learning and reverse engineering of colloidal self-assembly

An overwhelming diversity of colloidal building blocks with distinct sizes, materials and tunable interaction potentials are now available for colloidal self-assembly. The application space for materials composed of these building blocks is vast. To make progress in the rational design of new self-assembled materials, it is desirable to guide the experimental synthesis efforts by computational modelling. Here, we discuss computer simulation methods and strategies used for the design of soft materials created through bottom-up self-assembly of colloids and nanoparticles. We describe simulation techniques for investigating the self-assembly behaviour of colloidal suspensions, including crystal structure prediction methods, phase diagram calculations and enhanced sampling techniques, as well as their limitations. We also discuss the recent surge of interest in machine learning and reverse-engineering methods. Although their implementation in the colloidal realm is still in its infancy, we anticipate that these data-science tools offer new paradigms in understanding, predicting and (inverse) design of novel colloidal materials.

Effect of particle size distribution on polydisperse hard disks

Using Monte Carlo simulations, we systematically investigate the effect of particle size distribution on the phase behavior of polydisperse hard disks. Compared with the commonly used Gaussian-like polydisperse hard disks [P. Sampedro Ruiz, Q.-l. Lei, and R. Ni, Commun. Phys. 2, 70 (2019)], we find that the phase behavior of polydisperse hard-disk systems with lognormal and triangle distributions is significantly different. In polydisperse hard-disk systems of lognormal distributions, although the phase diagram appears similar to that of Gaussian-like polydisperse hard disks, the re-entrant melting of the hexatic or solid phase cannot be observed in sedimentation experiments. For polydisperse hard-disk systems of triangle distributions, the phase behavior is qualitatively different from the Gaussian-like and lognormal distributions, and we cannot reach any system of true polydispersity larger than 0.06, which is due to the special shape of the triangle distribution. Our results suggest that the exact particle size distribution is of primary importance in determining the phase behavior of polydisperse hard disks, and we do not have a universal phase diagram for different polydisperse hard-disk systems.

Parallel Prefetching for Canonical Ensemble Monte Carlo Simulations

In order to enable large-scale molecular simulations, algorithms must efficiently utilize multicore processors that continue to increase in total core count over time with relatively stagnant clock speeds. Although parallelized molecular dynamics (MD) software has taken advantage of this trend in computer hardware, single-particle perturbations with Monte Carlo (MC) are more difficult to parallelize than system-wide updates in MD using domain decomposition. Instead, prefetching reconstructs the serial Markov chain after computing multiple MC trials in parallel. Canonical ensemble MC simulations of a Lennard-Jones fluid with prefetching resulted in up to a factor of 1.7 speedup using 2 threads, and a factor of 3 speedup using 4 threads. Strategies for maximizing efficiency of prefetching simulations are discussed, including the potentially counterintuitive benefit of reduced acceptance probabilities. Determination of the optimal acceptance probability for a parallel simulation is simplified by theoretical prediction from serial simulation data. Finally, complete open-source code for parallel prefetch simulations was made available in the Free Energy and Advance Sampling Simulation Toolkit (FEASST).

Chaining of hard disks in nematic needles: particle-based simulation of colloidal interactions in liquid crystals

Colloidal particles suspended in liquid crystals can exhibit various effective anisotropic interactions that can be tuned and utilized in self-assembly processes. We simulate a two-dimensional system of hard disks suspended in a solution of dense hard needles as a model system for colloids suspended in a nematic lyotropic liquid crystal. The novel event-chain Monte Carlo technique enables us to directly measure colloidal interactions in a microscopic simulation with explicit liquid crystal particles in the dense nematic phase. We find a directional short-range attraction for disks along the director, which triggers chaining parallel to the director and seemingly contradicts the standard liquid crystal field theory result of a quadrupolar attraction with a preferred [Formula: see text] angle. Our results can be explained by a short-range density-dependent depletion interaction, which has been neglected so far. Directionality and strength of the depletion interaction are caused by the weak planar anchoring of hard rods. The depletion attraction robustly dominates over the quadrupolar elastic attraction if disks come close. Self-assembly of many disks proceeds via intermediate chaining, which demonstrates that in lyotropic liquid crystal colloids depletion interactions play an important role in structure formation processes.

Efficient Irreversible Monte Carlo Samplers

We present here two irreversible Markov chain Monte Carlo algorithms for general discrete state systems. One of the algorithms is based on the random-scan Gibbs sampler for discrete states and the other on its improved version, the Metropolized-Gibbs sampler. The algorithms we present incorporate the lifting framework with skewed detailed balance condition and construct irreversible Markov chains that satisfy the balance condition. We have applied our algorithms to 1D 4-state Potts model. The integrated autocorrelation times for magnetization and energy density indicate a reduction of the dynamical scaling exponent from z ≈ 1 to z ≈ 1/2. In addition, we have generalized an irreversible Metropolis-Hastings algorithm with skewed detailed balance, initially introduced by Turitsyn et al. [ Physica D 2011, 240, 410] for the mean field Ising model, to be now readily applicable to classical spin systems in general; application to 1D 4-state Potts model indicate a square root reduction of the mixing time at high temperatures.

Event-chain Monte Carlo simulations of the liquid to solid transition of two-dimensional decagonal colloidal quasicrystals

In event-chain Monte Carlo simulations, we model colloidal particles in two dimensions that interact according to an isotropic short-ranged pair potential which supports the two typical length scales present in decagonal quasicrystals. We investigate the assembled structures as we vary the density and temperature. Our special interest is related to the transition from quasicrystal to liquid. In contrast to the KTHNY melting theory for quasicrystals which predicts an intermediate pentahedratic phase, we find a one-step first-order melting transition. However, we discover that the slow relaxation of phasonic flips, i.e. rearrangements of the particles due to additional degrees of freedom in quasicrystals, changes the positional correlation functions, to the extent that structures with long-range orientational correlations, but exponentially decaying positional correlations, are observed.

Event-chain Monte Carlo with factor fields

We study the dynamics of one-dimensional (1D) interacting particles simulated with the event-chain Monte Carlo algorithm (ECMC). We argue that previous versions of the algorithm suffer from a mismatch in the factor potential between different particle pairs (factors) and show that in 1D models, this mismatch is overcome by factor fields. ECMC with factor fields is motivated, in 1D, for the harmonic model, and validated for the Lennard-Jones model as well as for hard spheres. In 1D particle systems with short-range interactions, autocorrelation times generally scale with the second power of the system size for reversible Monte Carlo dynamics, and with its first power for regular ECMC and for molecular dynamics. We show, using simulations, that the autocorrelation time grows only with the square root of the system size for ECMC with factor fields. Mixing times, which bound the time to reach equilibrium from an arbitrary initial configuration, grow with the first power of the system size.

Dynamical Computation of the Density of States and Bayes Factors Using Nonequilibrium Importance Sampling

Nonequilibrium sampling is potentially much more versatile than its equilibrium counterpart, but it comes with challenges because the invariant distribution is not typically known when the dynamics breaks detailed balance. Here, we derive a generic importance sampling technique that leverages the statistical power of configurations transported by nonequilibrium trajectories and can be used to compute averages with respect to arbitrary target distributions. As a dissipative reweighting scheme, the method can be viewed in relation to the annealed importance sampling (AIS) method and the related Jarzynski equality. Unlike AIS, our approach gives an unbiased estimator, with a provably lower variance than directly estimating the average of an observable. We also establish a direct relation between a dynamical quantity, the dissipation, and the volume of phase space, from which we can compute quantities such as the density of states and Bayes factors. We illustrate the properties of estimators relying on this sampling technique in the context of density of state calculations, showing that it scales favorable with dimensionality-in particular, we show that it can be used to compute the phase diagram of the mean-field Ising model from a single nonequilibrium trajectory. We also demonstrate the robustness and efficiency of the approach with an application to a Bayesian model comparison problem of the type encountered in astrophysics and machine learning.

Clock Monte Carlo methods

We propose the clock Monte Carlo technique for sampling each successive chain step in constant time. It is built on a recently proposed factorized transition filter and its core features include its O(1) computational complexity and its generality. We elaborate how it leads to the clock factorized Metropolis (clock FMet) method, and discuss its application in other update schemes. By grouping interaction terms into boxes of tunable sizes, we further formulate a variant of the clock FMet algorithm, with the limiting case of a single box reducing to the standard Metropolis method. A theoretical analysis shows that an overall acceleration of O(N^{κ}) (0≤κ≤1) can be achieved compared to the Metropolis method, where N is the system size and the κ value depends on the nature of the energy extensivity. As a systematic test, we simulate long-range O(n) spin models in a wide parameter regime: for n=1,2,3, with disordered, algebraically decaying or oscillatory Ruderman-Kittel-Kasuya-Yosida-type interactions and with and without external fields, and in spatial dimensions from d=1,2,3 to the mean field. The O(1) computational complexity is demonstrated, and the expected acceleration is confirmed. Its flexibility and its independence from the interaction range guarantee that the clock method would find decisive applications in systems with many interaction terms.

First-order and continuous melting transitions in two-dimensional Lennard-Jones systems and repulsive disks

In two-dimensional Lennard-Jones (LJ) systems, a small interval of melting-mode switching occurs below which the melting occurs by first-order phase transitions in lieu of the melting scenario proposed by Kosterlitz, Thouless, Halperin, Nelson, and Young (KTHNY). The extrapolated upper bound for phase coexistence is at density ρ∼0.893 and temperature T∼1.1, both in reduced LJ units. The two-stage KTHNY scenario is restored at higher temperatures, and the isothermal melting scenario is universal. The solid-hexatic and hexatic-liquid transitions in KTHNY theory, even so continuous, are distinct from typical continuous phase transitions in that instead of scale-free fluctuations, they are characterized by unbinding of topological defects, resulting in a special form of divergence of the correlation length: ξ≈exp(b|T-T_{c}|^{-ν}). Here such a divergence is firmly established for a two-dimensional melting phenomenon, providing a conclusive proof of the KTHNY melting. We explicitly confirm that this high-temperature melting behavior of the LJ system is consistent with the melting behavior of the r^{-12} potential and that melting of the r^{-n} potential is KTHNY-like for n≤12 but melting of the r^{-64} potential is first order; similar to hard disks. Therefore we suggest that the melting scenario of these repulsive potentials becomes hard-disk-like for an exponent in the range 12<n<64.

Computational construction of the electronic Hamiltonian for photoinduced electron transfer and Redfield propagation

The construction of open-system diabatic Hamiltonians relevant for the investigation of electron transfer processes is a computational challenge. Here all relevant parameters for Redfield propagations are extracted fromab initiocomputations.

The race to the bottom: approaching the ideal glass?

Key to resolving the scientific challenge of the glass transition is to understand the origin of the massive increase in viscosity of liquids cooled below their melting temperature (avoiding crystallisation). A number of competing and often mutually exclusive theoretical approaches have been advanced to describe this phenomenon. Some posit a bona fide thermodynamic phase to an 'ideal glass', an amorphous state with exceptionally low entropy. Other approaches are built around the concept of the glass transition as a primarily dynamic phenomenon. These fundamentally different interpretations give equally good descriptions of the data available, so it is hard to determine which-if any-is correct. Recently however this situation has begun to change. A consensus has emerged that one powerful means to resolve this longstanding question is to approach the putative thermodynamic transition sufficiently closely, and a number of techniques have emerged to meet this challenge. Here we review the results of some of these new techniques and discuss the implications for the existence-or otherwise-of the thermodynamic transition to an ideal glass.

Communication: Contact values of pair distribution functions in colloidal hard disks by test-particle insertion

We apply Henderson's method for measuring the cavity distribution function y(r) [J. Henderson, Mol. Phys. 48, 389 (1983)] to obtain the pair distribution function at contact, g(σ⁺). In contrast to the conventional distance-histogram method, no approximate extrapolation to contact is required. The resulting equation of state from experiments and simulations of hard disks agrees well with the scaled particle theory prediction up to high fluid packing fractions. We also provide the first experimental measurement of y(r) inside the hard core, which will allow for a more complete comparison with theory. The method's flexibility is further illustrated by measuring the partial pair distribution functions of binary hard-disk mixtures in simulation. The equation for the contact values can be used to derive familiar results from statistical geometry.

Clustering and assembly dynamics of a one-dimensional microphase former

Both ordered and disordered microphases ubiquitously form in suspensions of particles that interact through competing short-range attraction and long-range repulsion (SALR). While ordered microphases are more appealing materials targets, understanding the rich structural and dynamical properties of their disordered counterparts is essential to controlling their mesoscale assembly. Here, we study the disordered regime of a one-dimensional (1D) SALR model, whose simplicity enables detailed analysis by transfer matrices and Monte Carlo simulations. We first characterize the signature of the clustering process on macroscopic observables, and then assess the equilibration dynamics of various simulation algorithms. We notably find that cluster moves markedly accelerate the mixing time, but that event chains are of limited help in the clustering regime. These insights will inspire further study of three-dimensional microphase formers.

Irreversible Local Markov Chains with Rapid Convergence towards Equilibrium

We study the continuous one-dimensional hard-sphere model and present irreversible local Markov chains that mix on faster time scales than the reversible heat bath or Metropolis algorithms. The mixing time scales appear to fall into two distinct universality classes, both faster than for reversible local Markov chains. The event-chain algorithm, the infinitesimal limit of one of these Markov chains, belongs to the class presenting the fastest decay. For the lattice-gas limit of the hard-sphere model, reversible local Markov chains correspond to the symmetric simple exclusion process (SEP) with periodic boundary conditions. The two universality classes for irreversible Markov chains are realized by the totally asymmetric SEP (TASEP), and by a faster variant (lifted TASEP) that we propose here. We discuss how our irreversible hard-sphere Markov chains generalize to arbitrary repulsive pair interactions and carry over to higher dimensions through the concept of lifted Markov chains and the recently introduced factorized Metropolis acceptance rule.

Disappearance of the Hexatic Phase in a Binary Mixture of Hard Disks

Recent studies of melting in hard disks have confirmed the existence of a hexatic phase occurring in a narrow window of density which is separated from the isotropic liquid phase by a first-order transition, and from the solid phase by a continuous transition. However, little is known concerning the melting scenario in mixtures of hard disks. Here we employ tailored Monte Carlo simulations to elucidate the phase behavior of a system of large (l) and small (s) disks with diameter ratio σ_{l}/σ_{s}=1.4. We find that as small disks are introduced to a system of large ones, the stability window of the hexatic phase shrinks progressively until the line of continuous transitions terminates at an end point beyond which melting becomes a first-order liquid-solid transition. This occurs at surprisingly low concentrations of the small disks, c≲1%, emphasizing the fragility of the hexatic phase. We speculate that the change to the melting scenario is a consequence of strong fractionation effects, the nature of which we elucidate.

Population annealing simulations of a binary hard-sphere mixture

Population annealing is a sequential Monte Carlo scheme well suited to simulating equilibrium states of systems with rough free energy landscapes. Here we use population annealing to study a binary mixture of hard spheres. Population annealing is a parallel version of simulated annealing with an extra resampling step that ensures that a population of replicas of the system represents the equilibrium ensemble at every packing fraction in an annealing schedule. The algorithm and its equilibration properties are described, and results are presented for a glass-forming fluid composed of a 50/50 mixture of hard spheres with diameter ratio of 1.4:1. For this system, we obtain precise results for the equation of state in the glassy regime up to packing fractions φ≈0.60 and study deviations from the Boublik-Mansoori-Carnahan-Starling-Leland equation of state. For higher packing fractions, the algorithm falls out of equilibrium and a free volume fit predicts jamming at packing fraction φ≈0.667. We conclude that population annealing is an effective tool for studying equilibrium glassy fluids and the jamming transition.

Cell-veto Monte Carlo algorithm for long-range systems

We present a rigorous efficient event-chain Monte Carlo algorithm for long-range interacting particle systems. Using a cell-veto scheme within the factorized Metropolis algorithm, we compute each single-particle move with a fixed number of operations. For slowly decaying potentials such as Coulomb interactions, screening line charges allow us to take into account periodic boundary conditions. We discuss the performance of the cell-veto Monte Carlo algorithm for general inverse-power-law potentials, and illustrate how it provides a new outlook on one of the prominent bottlenecks in large-scale atomistic Monte Carlo simulations.

Eigenvalue analysis of an irreversible random walk with skew detailed balance conditions

An irreversible Markov-chain Monte Carlo (MCMC) algorithm with skew detailed balance conditions originally proposed by Turitsyn et al. is extended to general discrete systems on the basis of the Metropolis-Hastings scheme. To evaluate the efficiency of our proposed method, the relaxation dynamics of the slowest mode and the asymptotic variance are studied analytically in a random walk on one dimension. It is found that the performance in irreversible MCMC methods violating the detailed balance condition is improved by appropriately choosing parameters in the algorithm.

Event-chain algorithm for the Heisenberg model: Evidence for z≃1 dynamic scaling

We apply the event-chain Monte Carlo algorithm to the three-dimensional ferromagnetic Heisenberg model. The algorithm is rejection-free and also realizes an irreversible Markov chain that satisfies global balance. The autocorrelation functions of the magnetic susceptibility and the energy indicate a dynamical critical exponent z≈1 at the critical temperature, while that of the magnetization does not measure the performance of the algorithm. We show that the event-chain Monte Carlo algorithm substantially reduces the dynamical critical exponent from the conventional value of z≃2.

Monte Carlo simulation of dense polymer melts using event chain algorithms

We propose an efficient Monte Carlo algorithm for the off-lattice simulation of dense hard sphere polymer melts using cluster moves, called event chains, which allow for a rejection-free treatment of the excluded volume. Event chains also allow for an efficient preparation of initial configurations in polymer melts. We parallelize the event chain Monte Carlo algorithm to further increase simulation speeds and suggest additional local topology-changing moves ("swap" moves) to accelerate equilibration. By comparison with other Monte Carlo and molecular dynamics simulations, we verify that the event chain algorithm reproduces the correct equilibrium behavior of polymer chains in the melt. By comparing intrapolymer diffusion time scales, we show that event chain Monte Carlo algorithms can achieve simulation speeds comparable to optimized molecular dynamics simulations. The event chain Monte Carlo algorithm exhibits Rouse dynamics on short time scales. In the absence of swap moves, we find reptation dynamics on intermediate time scales for long chains.

Two-dimensional melting: from liquid-hexatic coexistence to continuous transitions

The phase diagram of two-dimensional continuous particle systems is studied using the event-chain Monte Carlo algorithm. For soft disks with repulsive power-law interactions ∝r^{-n} with n≳6, the recently established hard-disk melting scenario (n→∞) holds: a first-order liquid-hexatic and a continuous hexatic-solid transition are identified. Close to n=6, the coexisting liquid exhibits very long orientational correlations, and positional correlations in the hexatic are extremely short. For n≲6, the liquid-hexatic transition is continuous, with correlations consistent with the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) scenario. To illustrate the generality of these results, we demonstrate that Yukawa particles likewise may follow either the KTHNY or the hard-disk melting scenario, depending on the Debye-Hückel screening length as well as on the temperature.