arXiv subjects

Gabriel Stoltz

Publications and source records attributed to Gabriel Stoltz.

At least 73 records · Page 4Linked to original sources

arXiv2018

Longtime convergence of the Temperature-Accelerated Molecular Dynamics Method

The equations of the temperature-accelerated molecular dynamics (TAMD) method for the calculations of free energies and partition functions are analyzed. Specifically, the exponential convergence of the law of these stochastic processes is established, with a convergence rate close to the one of the limiting, effective dynamics at higher temperature obtained with infinite acceleration. It is also shown that the invariant measures of TAMD are close to a known reference measure, with an error that can be quantified precisely. Finally, a Central Limit Theorem is proven, which allows the estimation of errors on properties calculated by ergodic time averages. These results not only demonstrate the usefulness and validity range of the TAMD equations, but they also permit in principle to adjust the parameter in these equations to optimize their efficiency.

math.PR↗

arXiv2018

Spectral methods for Langevin dynamics and associated error estimates

We prove the consistency of Galerkin methods to solve Poisson equations where the differential operator under consideration is the generator of the Langevin dynamics. We show in particular how the hypocoercive nature of this operator can be used at the discrete level to first prove the invertibility of the rigidity matrix, and next provide error bounds on the approximation of the solution of the Poisson equation. We present general convergence results in an abstract setting, as well as explicit convergence rates for a simple one-dimensional example discretized using a tensor basis. Our theoretical findings are illustrated by numerical simulations.

math.NA↗

arXiv2017

Stable and accurate schemes for smoothed dissipative particle dynamics

Smoothed Dissipative Particle Dynamics (SDPD) is a mesoscopic particle method which allows to select the level of resolution at which a fluid is simulated. The numerical integration of its equations of motion still suffers from the lack of numerical schemes satisfying all the desired properties: energy conservation, parallelizability and stability. The similarities between SDPD and Dissipative Particle Dynamics with Energy conservation (DPDE), which is another coarse-grained model, enable the adaptation of recent numerical schemes developed for DPDE to the SDPD setting. In this article, we introduce a Metropolis step in the integration of the fluctuation/dissipation part of SDPD to improve its stability.

cond-mat.stat-mech↗

arXiv2017

Convergence and efficiency of adaptive importance sampling techniques with partial biasing

We consider a generalization of the discrete-time Self Healing Umbrella Sampling method, which is an adaptive importance technique useful to sample multimodal target distributions. The importance function is based on the weights (namely the relative probabilities) of disjoint sets which form a partition of the space. These weights are unknown but are learnt on the fly yielding an adaptive algorithm. In the context of computational statistical physics, the logarithm of these weights is, up to a multiplicative constant, the free energy, and the discrete valued function defining the partition is called the collective variable. The algorithm falls into the general class of Wang-Landau type methods, and is a generalization of the original Self Healing Umbrella Sampling method in two ways: (i) the updating strategy leads to a larger penalization strength of already visited sets in order to escape more quickly from metastable states, and (ii) the target distribution is biased using only a fraction of the free energy, in order to increase the effective sample size and reduce the variance of importance sampling estimators. The algorithm can also be seen as a generalization of well-tempered metadynamics. We prove the convergence of the algorithm and analyze numerically its efficiency on a toy example.

math.PR↗

arXiv2017

Convergence rates for nonequilibrium Langevin dynamics

We study the exponential convergence to the stationary state for nonequilibrium Langevin dynamics, by a perturbative approach based on hypocoercive techniques developed for equilibrium Langevin dynamics. The Hamiltonian and overdamped limits (corresponding respectively to frictions going to zero or infinity) are carefully investigated. In particular, the maximal magnitude of admissible perturbations are quantified as a function of the friction. Numerical results based on a Galerkin discretization of the generator of the dynamics confirm the theoretical lower bounds on the spectral gap.

math-ph↗

arXiv2017

Stable schemes for dissipative particle dynamics with conserved energy

This article presents a new numerical scheme for the discretization of dissipative particle dynamics with conserved energy. The key idea is to reduce elementary pairwise stochastic dynamics (either fluctuation/dissipation or thermal conduction) to effective single-variable dynamics, and to approximate the solution of these dynamics with one step of a Metropolis-Hastings algorithm. This ensures by construction that no negative internal energies are encountered during the simulation, and hence allows to increase the admissible timesteps to integrate the dynamics, even for systems with small heat capacities. Stability is only limited by the Hamiltonian part of the dynamics, which suggests resorting to multiple timestep strategies where the stochastic part is integrated less frequently than the Hamiltonian one.

cond-mat.stat-mech↗

arXiv2017

Robust determination of maximally-localized Wannier functions

We propose an algorithm to determine Maximally Localized Wannier Functions (MLWFs). This algorithm, based on recent theoretical developments, does not require any physical input such as initial guesses for the Wannier functions, unlike popular schemes based on the projection method. We discuss how the projection method can fail on fine grids when the initial guesses are too far from MLWFs. We demonstrate that our algorithm is able to find localized Wannier functions through tests on two-dimensional systems, simplified models of semiconductors, and realistic DFT systems by interfacing with the Wannier90 code. We also test our algorithm on the Haldane and Kane-Mele models to examine how it fails in the presence of topological obstructions.

cond-mat.mtrl-sci↗

arXiv2016

An accurate scheme to solve cluster dynamics equations using a Fokker-Planck approach

We present a numerical method to accurately simulate particle size distributions within the formalism of rate equation cluster dynamics. This method is based on a discretization of the associated Fokker-Planck equation. We show that particular care has to be taken to discretize the advection part of the Fokker-Planck equation, in order to avoid distortions of the distribution due to numerical diffusion. For this purpose we use the Kurganov-Noelle-Petrova scheme coupled with the monotonicity-preserving reconstruction MP5, which leads to very accurate results. The interest of the method is highlighted on the case of loop coarsening in aluminum. We show that the choice of the models to describe the energetics of loops does not significantly change the normalized loop distribution, while the choice of the models for the absorption coefficients seems to have a significant impact on it.

cond-mat.mtrl-sci↗

arXiv2016

Cluster dynamics modelling of materials: a new hybrid deterministic/stochastic coupling approach

Deterministic simulations of the rate equations governing cluster dynamics in materials are limited by the number of equations to integrate. Stochastic simulations are limited by the high frequency of certain events. We propose a coupling method combining deterministic and stochastic approaches. It allows handling different time scale phenomena for cluster dynamics. This method, based on a splitting of the dynamics, is generic and we highlight two different hybrid deterministic/stochastic methods. These coupling schemes are highly parallelizable and specifically designed to treat large size cluster problems. The proof of concept is made on a simple model of vacancy clustering under thermal ageing.

cond-mat.mtrl-sci↗

arXiv2016

Improving dynamical properties of metropolized discretizations of overdamped Langevin dynamics

The discretization of overdamped Langevin dynamics, through schemes such as the Euler-Maruyama method, can be corrected by some acceptance/rejection rule, based on a Metropolis-Hastings criterion for instance. In this case, the invariant measure sampled by the Markov chain is exactly the Boltzmann-Gibbs measure. However, rejections perturb the dynamical consistency of the resulting numerical method with the reference dynamics. We present in this work some modifications of the standard correction of discretizations of overdamped Langevin dynamics on compact spaces by a Metropolis-Hastings procedure, which allow us to either improve the strong order of the numerical method, or to decrease the bias in the estimation of transport coefficients characterizing the effective dynamical behavior of the dynamics. For the latter approach, we rely on modified numerical schemes together with a Barker rule for the acceptance/rejection criterion.

math.NA↗

arXiv2016

Error Analysis of Modified Langevin Dynamics

We consider Langevin dynamics associated with a modified kinetic energy vanishing for small momenta. This allows us to freeze slow particles, and hence avoid the re-computation of inter-particle forces, which leads to computational gains. On the other hand, the statistical error may increase since there are a priori more correlations in time. The aim of this work is first to prove the ergodicity of the modified Langevin dynamics (which fails to be hypoelliptic), and next to analyze how the asymptotic variance on ergodic averages depends on the parameters of the modified kinetic energy. Numerical results illustrate the approach, both for low-dimensional systems where we resort to a Galerkin approximation of the generator, and for more realistic systems using Monte Carlo simulations.

cond-mat.stat-mech↗

arXiv2015

Coupling a reactive potential with a harmonic approximation for atomistic simulations of material failure

Molecular dynamics (MD) simulations involving reactive potentials can be used to model material failure. The empirical potentials which are used in such simulations are able to adapt to the atomic environment, at the expense of a significantly higher computational cost than non-reactive potentials. However, during a simulation of failure, the reactive ability is needed only in some limited parts of the system, where bonds break or form and the atomic environment changes. Therefore, simpler non-reactive potentials can be used in the remainder of the system, provided that such potentials reproduce correctly the behavior of the reactive potentials in this region, and that seamless coupling is ensured at the interface between the reactive and non-reactive regions. In this article, we propose a methodology to combine a reactive potential with a non-reactive approximation thereof, made of a set of harmonic pair and angle interactions and whose parameters are adjusted to predict the same energy, geometry and Hessian in the ground state of the potential. We present a methodology to construct the non-reactive approximation of the reactive potential, and a way to couple these two potentials. We also propose a criterion for on-the-fly substitution of the reactive potential by its non-reactive approximation during a simulation. We illustrate the correctness of this hybrid technique for the case of MD simulation of failure in two-dimensional graphene originally modeled with REBO potential.

cond-mat.mtrl-sci↗

arXiv2015

A mathematical analysis of the GW0 method for computing electronic excited energies of molecules

This paper analyses the GW method for finite electronic systems. In a first step, we provide a mathematical framework for the usual one-body operators that appear naturally in many-body perturbation theory. We then discuss the GW equations which construct an approximation of the one-body Green's function, and give a rigorous mathematical formulation of these equations. Finally, we study the well-posedness of the GW0 equations, proving the existence of a unique solution to these equations in a perturbative regime.

math-ph↗

arXiv2015

Permutation-invariant distance between atomic configurations

We present a permutation-invariant distance between atomic configurations, defined through a functional representation of atomic positions. This distance enables to directly compare different atomic environments with an arbitrary number of particles, without going through a space of reduced dimensionality (i.e. fingerprints) as an intermediate step. Moreover, this distance is naturally invariant through permutations of atoms, avoiding the time consuming associated minimization required by other common criteria (like the Root Mean Square Distance). Finally, the invariance through global rotations is accounted for by a minimization procedure in the space of rotations solved by Monte Carlo simulated annealing. A formal framework is also introduced, showing that the distance we propose verifies the property of a metric on the space of atomic configurations. Two examples of applications are proposed. The first one consists in evaluating faithfulness of some fingerprints (or descriptors), i.e. their capacity to represent the structural information of a configuration. The second application concerns structural analysis, where our distance proves to be efficient in discriminating different local structures and even classifying their degree of similarity.

cond-mat.mtrl-sci↗

arXiv2015

The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics

We consider numerical methods for thermodynamic sampling, i.e. computing sequences of points distributed according to the Gibbs-Boltzmann distribution, using Langevin dynamics and overdamped Langevin dynamics (Brownian dynamics). A wide variety of numerical methods for Langevin dynamics may be constructed based on splitting the stochastic differential equations into various component parts, each of which may be propagated exactly in the sense of distributions. Each such method may be viewed as generating samples according to an associated invariant measure that differs from the exact canonical invariant measure by a stepsize-dependent perturbation. We provide error estimates a la Talay-Tubaro on the invariant distribution for small stepsize, and compare the sampling bias obtained for various choices of splitting method. We further investigate the overdamped limit and apply the methods in the context of driven systems where the goal is sampling with respect to a nonequilibrium steady state. Our analyses are illustrated by numerical experiments.

cond-mat.stat-mech↗

arXiv2014

Nonlinear Fluctuating Hydrodynamics in One Dimension: the Case of Two Conserved Fields

We study the BS model, which is a one-dimensional lattice field theory taking real values. Its dynamics is governed by coupled differential equations plus random nearest neighbor exchanges. The BS model has exactly two locally conserved fields. Through numerical simulations the peak structure of the steady state space-time correlations is determined and compared with nonlinear fluctuating hydrodynamics, which predicts a traveling peak with KPZ scaling function and a standing peak with a scaling function given by the completely asymmetric Levy distribution with parameter $α= 5/3$. As a by-product, we completely classify the universality classes for two coupled stochastic Burgers equations with arbitrary coupling coefficients.

cond-mat.stat-mech↗

arXiv2014

Free energy calculations from adaptive molecular dynamics simulations with adiabatic reweighting

We propose an adiabatic reweighting algorithm for computing the free energy along an external parameter from adaptive molecular dynamics simulations. The adaptive bias is estimated using Bayes identity and information from all the sampled configurations. We apply the algorithm to a structural transition in a cluster and to the migration of a crystalline defect along a reaction coordinate. Compared to standard adaptive molecular dynamics, we observe an acceleration of convergence. With the aid of the algorithm, it is also possible to iteratively construct the free energy along the reaction coordinate without having to differentiate the gradient of the reaction coordinate or any biasing potential.

cond-mat.stat-mech↗

arXiv2013

Local density dependent potential for compressible mesoparticles

We focus on finding a coarse grained description able to reproduce the thermodynamic behavior of a molecular system by using mesoparticles representing several molecules. Interactions between mesoparticles are modelled by an interparticle potential, and an additional internal equation of state is used to account for the thermic contribution of coarse grained internal degrees of freedom. Moreover, as strong non-equilibrium situations over a wide range of pressure and density are targeted, the internal compressibility of these mesoparticles has to be considered. This is done by introducing a dependence of the potential on the local environment of the mesoparticles, either by defining a spherical local density or by means of a Voronoi tessellation. As an example, a local density dependent potential is fitted to reproduce the Hugoniot curve of a model of nitromethane, where each mesoparticle represents one thousand molecules.

cond-mat.soft↗