arXiv subjects

Pierre Terrier

Publications and source records attributed to Pierre Terrier.

4 recordsLinked to original sources

arXiv2019

A mathematical justification of the finite time approximation of Becker-Döring equations by a Fokker-Planck dynamics

The Becker-Döring equations are an infinite dimensional system of ordinary differntial equations describing coagulation/fragmentation processes of species of integer sizes. Formal Taylor expansions motivate that its solution should be well described by a partial differential equation for large sizes, of advection-diffusion type, called Fokker-Planck equation. We rigorously prove the link between these two descriptions for evolutions on finite times rather than in some hydrodynamic limit, motivated by the results of numerical simulations and the construction of dedicated algorithms based on splitting strategies. In fact, the Becker-Döring equations and the Fokker-Planck equation are related through some pure diffusion with unbounded diffusion coefficient. The crucial point in the analysis is to obtain decay estimates for the solution of this pure diffusion and its derivates to control remainders in the Taylor expansions. The small parameter in this analysis is the inverse of the minimal size of the species.

math.CA↗

arXiv2017

Estimating thermodynamic expectations and free energies in expanded ensemble simulations: systematic variance reduction through conditioning

Markov chain Monte Carlo methods are primarily used for sampling from a given probability distribution and estimating multi-dimensional integrals based on the information contained in the generated samples. Whenever it is possible, more accurate estimates are obtained by combining Monte Carlo integration and integration by numerical quadrature along particular coordinates. We show that this variance reduction technique, referred to as conditioning in probability theory, can be advantageously implemented in \emph{expanded ensemble} simulations. These simulations aim at estimating thermodynamic expectations as a function of an external parameter that is sampled like an additional coordinate. Conditioning therein entails integrating along the external coordinate by numerical quadrature. We prove variance reduction with respect to alternative standard estimators and demonstrate the practical efficiency of the technique by estimating free energies and characterizing a structural phase transition between two solid phases.

cond-mat.stat-mech↗

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↗

arXiv2015

Using Bayes formula to estimate rates of rare events in transition path sampling simulations

Transition path sampling is a method for estimating the rates of rare events in molecular systems based on the gradual transformation of a path distribution containing a small fraction of reactive trajectories into a biased distribution in which these rare trajectories have become frequent. Then, a multistate reweighting scheme is implemented to postprocess data collected from the staged simulations. Herein, we show how Bayes formula allows to directly construct a biased sample containing an enhanced fraction of reactive trajectories and to concomitantly estimate the transition rate from this sample. The approach can remediate the convergence issues encountered in free energy perturbation or umbrella sampling simulations when the transformed distribution insufficiently overlaps with the reference distribution.

cond-mat.stat-mech↗