Mathematical Foundations of Accelerated Molecular Dynamics Methods

Lelièvre, Tony

doi:10.1007/978-3-319-42913-7_27-1

Cited by 8 publications

(11 citation statements)

References 41 publications

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“…In the context of statistical physics, this behavior is expected since the molecular system typically jumps between various conformations, which are indeed these metastable states. For modeling purposes as well as for building efficient numerical methods (see for instance [1][2][3]), it is thus interesting to be able to precisely describe the exit event from a metastable state, namely the law of the first exit time and the first exit point.…”

Section: Setting and Motivationmentioning

confidence: 99%

“…Using partial differential equation techniques, some of the formal results above have been rigorously proven. For example, when ∂ n f > 0 on ∂Ω, (3) and {x ∈ Ω, |∇f (x)| = 0} = {x 0 } with f (x 0 ) = min Ω f and det Hessf (x 0 ) > 0, (4) the concentration of the law of X τ Ω in the limit h → 0 on arg min ∂Ω f has been obtained in [8][9][10], when X 0 = x ∈ Ω, see also [11,12] for more recent results with similar techniques. Finally, another rigorous approach to study the exit point distribution is to rely on the theory of large deviations.…”

Section: Setting and Motivationmentioning

confidence: 99%

“…Let us emphasize that this work requires that X 0 is distributed according to the quasi-stationary distribution ν h in Ω (see Definition 2 below). This is relevant when the exit from the domain Ω is metastable, namely when the process (1) reaches a local equilibrium within Ω before exiting Ω (see Section 1.2 below and [1][2][3]). The companion paper [18] builds on and extends the result of the present work to general initial conditions in Ω, as explained in Remark 8 below.…”

Section: Setting and Motivationmentioning

confidence: 99%

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The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 2

Lelièvre

Peutrec

Nectoux

2021

Stoch PDE: Anal Comp

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We consider the first exit point distribution from a bounded domain Ω of the stochastic process (X t ) t≥0 solution to the overdamped Langevin dynamicsstarting from the quasi-stationary distribution in Ω. In the small temperature regime (h → 0) and under rather general assumptions on f (in particular, f may have several critical points in Ω), it is proven that the support of the distribution of the first exit point concentrates on some points realizing the minimum of f on ∂Ω. Some estimates on the relative likelihood of these points are provided. The proof relies on tools from semi-classical analysis.r é s u m é Dans ce travail, nous étudions la distribution du point de sortie d'un domaine borné Ω pour le processus stochastique (X t ) t≥0 solution de la dynamique de Langevin suramortieinitialement distribué suivant la distribution quasi-stationnaire dans Ω. Dans la limite basse température h → 0 et sous des hypothèses générales sur la fonction f (f pouvant notamment avoir plusieurs points critiques dans Ω), nous montrons que la distribution du lieu de sortie se concentre sur certains points réalisant le minimum de f sur ∂Ω. Nous calculons aussi les probabilités relatives de sortir autour de chacun de ces points. Nos preuves reposent sur des outils issus de l'analyse semi-classique.

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Section: Setting and Motivationmentioning

confidence: 99%

Section: Setting and Motivationmentioning

confidence: 99%

Section: Setting and Motivationmentioning

confidence: 99%

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The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 2

Lelièvre

Peutrec

Nectoux

2021

Stoch PDE: Anal Comp

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“…In the general case, atomistic mechanisms must be discovered through unbiased dynamic 10,[16][17][18][19][20] or static 11,21,22 sampling approaches. When the true dynamics can be characterized as rare transitions between metastable basins on the energy landscape 23 , the basin-to-basin dynamics can be mapped to a continuous-time Markov chain 24,25 , which forms the theoretical foundation of atomistic kinetic Monte Carlo (akMC) methods 26 . The resulting model can then be stochastically or in some cases (such as that presented here) analytically integrated to extract observables of interest.…”

Section: Introductionmentioning

confidence: 99%

Automated calculation and convergence of defect transport tensors

Swinburne

Pérez

2020

npj Comput Mater

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Defect diffusion is a key process in materials science and catalysis, but as migration mechanisms are often too complex to enumerate a priori, calculation of transport tensors typically have no measure of convergence and require significant end-user intervention. These two bottlenecks prevent high-throughput implementations essential to propagate model-form uncertainty from interatomic interactions to predictive simulations. In order to address these issues, we extend a massively parallel accelerated sampling scheme, autonomously controlled by Bayesian estimators of statewide sampling completeness, to build atomistic kinetic Monte Carlo models on a state-space irreducible under exchange and space group symmetries. Focusing on isolated defects, we derive analytic expressions for drift and diffusion coefficients, providing a convergence metric by calculating the Kullback–Leibler divergence across the ensemble of diffusion processes consistent with the sampling uncertainty. The autonomy and efficacy of the method is demonstrated on surface trimers in tungsten and Hexa-interstitials in magnesium oxide, both of which exhibit complex, correlated migration mechanisms.

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“…In the thermally activated regime, any system is extremely likely to thermalize in a local energy minima before escaping, giving a well defined separation of timescales between vibrations and transitions. In this limit, the atomic dynamics can be mapped to a continuous time, discrete state Markov chain 19 , which provides the theoretical basis of off-lattice, or atomistic kinetic Monte Carlo (akMC) methods 20 , up to an error exponentially small in the timescale separation 21 . We have employed this rigorous connection in our massively parallel sampling scheme TAMMBER 15,16,22 (available at github.com/tomswinburne/tammber), which optimally manages many thousands of molecular dynamics 'workers' to rapidly discover migration pathways of complex defects, with a novel Bayesian metric of sampling completeness which can be used to assign well defined uncertainty bounds on the resulting akMC model.…”

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confidence: 99%

Reaction-drift-diffusion models from master equations: application to material defects

Swinburne,

Perez

2021

Preprint

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We present a general method to produce well-conditioned continuum reaction-drift-diffusion equations directly from master equations on a discrete, periodic state space. We assume the underlying data to be kinetic Monte Carlo models (i.e., continuous-time Markov chains) produced from atomic sampling of point defects in locally periodic environments, such as perfect lattices, ordered surface structures or dislocation cores, possibly under the influence of a slowly varying external field. Our approach also applies to any discrete, periodic Markov chain. The analysis identifies a previously omitted non-equilibrium drift term, present even in the absence of external forces, which can compete in magnitude with the reaction rates, thus being essential to correctly capture the kinetics. To remove fast modes which hinder time integration, we use a generalized Bloch relation to efficiently calculate the eigenspectrum of the master equation. A well conditioned continuum equation then emerges by searching for spectral gaps in the long wavelength limit, using an established kinetic clustering algorithm (e.g., PCCA+) to define a proper reduced state space.

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Mathematical Foundations of Accelerated Molecular Dynamics Methods

Cited by 8 publications

References 41 publications

The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 2

The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 2

Automated calculation and convergence of defect transport tensors

Reaction-drift-diffusion models from master equations: application to material defects

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