Session C38: Advances in Computational Statistical Mechanics and their Applications: Part 3

Fast irreversible Markov chains in statistical mechanics

The Markov-chain Monte Carlo method is an outstanding computational tool in science. Since its origins, it has relied on the detailed-balance condition and the Metropolis algorithm to solve general computational problems under the condition of thermodynamic equilibrium with zero probability flows.
In this talk, I discuss a class of 'event-chain' algorithms that violate detailed balance yet satisfy global balance (the Markov chains are irreversible). Equilibrium is realized as a steady state with non-vanishing probability flows. The time-honored Metropolis acceptance criterion based on the change in the energy is replaced by a consensus rule originating in the new factorized Metropolis filter. The system energy is not computed, providing a fresh perspective for long-range interactions. Moves are infinitesimal and persistent, and they implement the lifting concept. The resulting general class of fast algorithms overcomes the limitation of detailed balance and also goes beyond hybrid Monte Carlo.
As applications I discuss our recent solution of the two-dimensional melting problem for hard disks and general potentials, and present the cell-veto algorithm for treating long-range systems without cutoffs nor Ewald summations. I will also discuss some open mathematical and algorithmic problems as well as connections of our algorithms with the totally asymmetric simple exclusion process (TASEP) and related processes.   

Rapid heterogeneous molecular simulation in time and space with parsemonious domain decomposition scheme

Multiscale and inhomogeneous molecular systems are challenging topics in the field of molecular simulation. In particular, mapping time and space of the different regions of a simulation with different resolution constitutes a new question worth exploring. Mainly, this question arises from the method development viewpoint for the domain decomposition of the molecular dynamics machinery. Here, we introduce the heterogeneous time-spatial domain decomposition approach which is a combination of an heterogeneity sensitive spatial domain decomposition with a time evolution average of particles' diffusion domainwise estimated. Within this approach, the spatial domain decomposition is theoretically modeled and results in scaling-laws for the force calculation simulation-time, while timewise the domains are tackled by the communication-time expended in the parallelization scheme. We explore the new approach capabilities, by comparing it to both static domain decomposition algorithms and dynamic load balancing schemes for archetypical molecular systems.   

World-line Quantum Monte Carlo Simulation of the Hubbard-Holstein Model

Quantum Monte Carlo (QMC) method is a powerful tool to understand the physics of strongly correlated interacting quantum system. Persistent questions concerning the role of phonons in strongly correlated materials like manganites, cuprates, iron pnictides, and organic superconductors keep the study of electron-phonon Hamiltonians at the forefront of research. We use world-line quantum Monte Carlo (WLQMC) to explore the ground state and finite temperature properties of one-dimensional many-body systems based on the extended Hubbard-Holstein Hamiltonian. This method is based on a direct-space, imaginary-time representation of the fermion and boson fields, which allows us to describe the regimes with dominant charge-density-wave, superconductivity and metallic behavior. We show the ground state and finite temperature phase diagrams of the Hubbard-Holstein model as a function of Hubbard U, nearest neighbor interaction V, phonon frequency ω and electron-phonon coupling constant λ. We show the simulation results for an extension of the Hamiltonian to a situation where electron-phonon interaction is long range.   

Critical nonequilibrium relaxation in cluster-update quantum Monte Carlo algorithms and its application to quantum phase transitions

Although improved quantum Monte Carlo (QMC) algorithms are based on the cluster-update scheme, previous applications of the nonequilibrium relaxation (NER) scheme to QMC calculations were based on the local-update scheme [1,2] because of “too fast relaxation of the cluster-update scheme for NER analyses”. Recently we found that the critical NER in cluster algorithms is described by the stretched-exponential simulation-time dependence, not the power-law one [3]. In the present study we analyze the Néel-dimer quantum phase transition of the S=1/2 columnar-dimerized antiferromagnetic Heisenberg model on a square lattice with the continuous-time loop algorithm and NER from the isolated dimer configuration. Our estimate of the critical point δ_c≈0.9095 (ratio of the strength of dimerized bonds to normal ones is (1+δ):1) is consistent with a recent QMC estimate δ_c≈0.90947(3) [4], and the relaxation exponent σ is consistent with that of the three-dimensional classical Heisenberg model, σ≈1/2 [5].

[1] Y. Nonomura, J. Phys. A: Math. Gen. 31, 7939 (1998).
[2] T. Nakamura and Y. Ito, J. Phys. Soc. Jpn. 72, 2405 (2003).
[3] Y. Nonomura, J. Phys. Soc. Jpn. 83, 113001 (2014).
[4] S. Yasuda and S. Todo, Phys. Rev. E 88, 061301(R) (2013).
[5] Y. Nonomura and Y. Tomita, Phys. Rev. E 93, 012101 (2016).   

Spatially Dependent Thermodynamic Integration: A Method to Compute Chemical Potentials of Dense Fluids and Concentrated Liquid Mixtures

Well established computational methods aiming at calculating chemical potentials rely on inserting test particles in the target system. The increase in density or concentration renders this procedure unfeasible, and the use of more sophisticated sampling techniques becomes inevitable. We propose an alternative strategy based on the Hamiltonian adaptive resolution framework. Here, the molecules of the target system, described with the appropriate resolution, are in physical contact with a reservoir of molecules modeled as ideal gas particles. To enforce a uniform density profile across the simulation box, a single-molecule external potential is computed, applied and identified with the excess chemical potential of the target system. We validate the method by computing chemical potentials of various molecular liquids, including aqueous solutions of sodium chloride.   

Forward-Flux Sampling with Jumpy Order Parameters

Forward-flux sampling (FFS) [1] is a path sampling technique that has gained popularity in recent years, and has been used to compute rates of rare-event driven phenomena such as crystallization, condensation, hydrophobic evaporation and protein folding. The popularity of FFS is not only due to its ease of implementation, but also because of its lack of sensitivity to the particular choice of an order parameter. The order parameter utilized in conventional FFS, however, still needs to satisfy a stringent smoothness criterion in order to assure sequential crossing of FFS milestones. This condition is usually violated for order parameters utilized for describing aggregation phenomena such as crystallization. Here, we present a generalized FFS algorithm for which this smoothness criterion is no longer necessary, and apply it to compute homogeneous crystal nucleation rates in several systems. Our numerical tests reveal that conventional FFS can underestimate rate by several orders of magnitude.

[1] Allen, et al., J. Chem. Phys., 124: 194111 (2006).
  

Gibbs-Sampling Enhanced Replica-Exchange Simulations

We recently introduced a novel replica-exchange scheme in which an individual replica can sample from states encountered by other replicas at any previous time by way of a global configuration database, enabling the fast propagation of relevant states through the whole ensemble of replicas. This mechanism depends on the knowledge of global thermodynamic functions which are measured during the simulation and not coupled to the heat bath temperatures driving the individual simulations. Therefore, this setup also allows for a continuous adaptation of the temperature set. In this talk, we will review the new scheme and demonstrate its capability. The method is particularly useful for the fast and reliable estimation of the microcanonical temperature T(U) or, equivalently, of the density of states g(U) over a wide range of energies.   

Accelerated histogram-free multicanonical Monte Carlo algorithm for the basis expansion of density of states

We propose a novel Monte Carlo algorithm to obtain the density of states (DOS) in the form of a basis expansion, for physical systems with continuous state variables [1]. Our algorithm inherits the strengths of various previous methods such as multicanonical sampling and Wang-Landau sampling. Additionally, we advance the capability of these methods: Firstly, instead of a numerical array of finite resolutions, our algorithm obtains a basis set expansion for the DOS, with the number of terms and coefficients determined and refined iteratively during the simulation. Secondly, since the visited energies are stored directly in a data set instead of a histogram, this reduces the undesirable statistical noise and errors caused by the discretization of state observables. Thirdly, as the random walkers are directed to achieve uniform sampling of the phase space, our scheme is more efficient and we have demonstrated an order of magnitude of speedup compared to existing methods. We will show how this method is applied to accelerate the simulation of materials properties.
[1] Y. W. Li and M. Eisenbach, in Proceedings of PASC ’17, ACM, New York, NY, USA, Article 10 (2017).   

Random number generators for the large-scale Monte-Carlo simulations

Monte Carlo simulations needs random numbers. History shows how problems with the quality of the PRN does biased results of Monte Carlo simulations. It happens with the higher scale of the simulations. Many improvements was made to make PRN generators more reliable and safe. Attention should be paid also for the possible influence of the correlations due to the high parallelism in the simulations. In the presentation we will present discussion of the problems and solutions for the massive parallel simulations on the clusters. We will present details of our approach and the corresponding libraries [1,2]. Examples will include generation of uncorrelated parallel streams of PRN using CPU, GPU, and CPU extensions SSE2, AVX2, and new AVX512 technology.

[1] Guskova M., Barash L. Y., L.N. Shchur. RNGAVXLIB: Program library for random number generation, AVX realization, Computer Physics Communications. 2016, vol 200, pp. 402-405.
[2] Barash L. Y., Shchur L. PRAND: GPU accelerated parallel random number generation library: Using most reliable algorithms and applying parallelism of modern GPUs and CPUs, Computer Physics Communications. 2014. vol. 185. pp. 1343-1353.   

Inverse Design of Pressure-Induced Solid—Solid Transitions in Colloids Using the Alchemical Ensemble

Recent developments in anisotropic particle synthesis have shown promise for using these particles as building blocks for functional materials. However, due to the large design spaces that are available to us, it can be challenging to find appropriate building blocks for target behaviors. Here, instead of mapping out the phase behavior for a range of particle shapes, we present a new, alternative computational statistical mechanical approach that couples together multiple systems within the “alchemical ensemble” — a generalized, statistical mechanical framework in which model attributes are allowed to fluctuate along with thermodynamic variables. The alchemical ensemble has application to a wide variety of design problems within statistical mechanics. Here, we demonstrate its use in the design of particle shapes for target materials properties, and present examples of candidate colloidal- and nano-particles designed within the alchemical ensemble specifically for structurally reconfigurable colloidal crystals.   

A random walk in function space: Statistical optimization of functional forms for physical models

For computer simulations of physical systems, it is necessary to obtain models that represent the underlying physics, while reducing the computational complexity required for its evaluation. Traditionally, this model construction has been a very time-consuming task. We present a method that allows us to automatically derive symbolic forms of model Hamiltonians from reference data that can be obtained from higher accuracy calculations, such as electronic structure methods and from experimental input. Our method is based on Gene Expression Programming (GEP) [1] techniques to sample the space of possible functional forms of classical Hamiltonians by statistical sampling of symbolic representations that guarantee well-formed expressions within a genetic algorithm. We compare the original GEP method to our two-step approach that separates functional and parameter optimization for test cases of function fitting and reproduction of pair-potentials and we show the performance improvements due to this decomposition of search spaces.

[1] C. Ferreira, Complex Systems 13, 87 (2001).   

Identification of Liquid-like and Gas-like Particles in Supercritical Fluid via Machine Learning Approach

Abrupt phase transition between vapor and liquid terminates at the liquid-gas critical point, where a single phase of supercritical fluid (SCF) emerges. In this work, adopting machine learning techniques, we propose a novel viewpoint that SCF is an inhomogeneous mixture of liquid-like and gas-like particles. We trained a neural network with local structure data of vapor and liquid particles, generated by molecular dynamics simulations in the near-critical condition; the neural network was trained to label individual particles as 'liquid-like' or 'gas-like'. The trained neural network was then used to identify the coexisting liquid-like and gas-like particles in SCF. The proportion of gas-like particles showed a well-defined dependence on bulk density, agreeing well with the prediction from two-state thermodynamics of interchangeable particles. From the distributions of liquid-like and gas-like particles, we present new explanations on some important properties of SCF, including the local density augmentation and divergent partial molar volume in supercritical solution.   

Computational Design of Microfabricated Thermionic Converters

Thermionic generators are a promising alternative to traditional mechanical generators for heat-to-electricity conversion. Here we discuss the use of computational studies for the design and analysis of high efficiency and high power density microfabricated thermionic converters. Our calculations use kinetic particle-in-cell electron dynamics simulations to capture non-equilibrium thermodynamic properties. We demonstrate quantitative agreement between predicted and experimentally measured properties of the thermionic devices, and discuss how simulations can be used to study device performance. This approach is widely applicable to vacuum electronic devices that incorporate charged particle optics.