Molecular dynamics and the accuracy of numerically computed averages

Abstract

Molecular dynamics is discussed from a mathematical perspective. The recent history of method development is briefly surveyed with an emphasis on the use of geometric integration as a guiding principle. The recovery of statistical mechanical averages from molecular dynamics is then introduced, and the use of backward error analysis as a technique for analysing the accuracy of numerical averages is described. This article gives the first rigorous estimates for the error in statistical averages computed from molecular dynamics simulation based on backward error analysis. It is shown that molecular dynamics introduces an appreciable bias at stepsizes which are below the stability threshold. Simulations performed in such a regime can be corrected by use of a stepsize-dependent reweighting factor. Numerical experiments illustrate the efficacy of this approach. In the final section, several open problems in dynamics-based molecular sampling are considered. CONTENTS

Equipartition and the Calculation of Temperature in Biomolecular Simulations

“…Finding H δ t is a problem of backward error analysis. 15,24 One may construct an asymptotic expansion for H δ t by adding terms to H 0 to create a Hamiltonian whose exact dynamics matches that of eqs 10 and 11 order by order in δ t . Since velocity Verlet is symmetric (reversing the sign of the time step gives the inverse method), only even powers of δ t appear It is possible to construct accurate numerical approximations for the correction terms; 25 we note that an estimator (eq 68 of ref 26) for δH (2) with errors of order δ t 2 (and thus an estimator of H δ t with errors of order δ t 4 ) has existed for many years in the CHARMM code, 27 where it is called a "highfrequency correction."…”

Variational collision integrator for polymer chains

“…While the discrete trajectory is close to the exact solution of a single modified Hamiltonian system away from collisions, the stability argument implied by backward error analysis is no longer applicable here. In [6], a modified collision Verlet algorithm is proposed that conserves the modified Hamiltonian (to fourth order accuracy using a higher order method) during the collision steps and thereby gains numerical stability. Note that collision times are determined solving a quartic equation in [6].…”

“…In [6], a modified collision Verlet algorithm is proposed that conserves the modified Hamiltonian (to fourth order accuracy using a higher order method) during the collision steps and thereby gains numerical stability. Note that collision times are determined solving a quartic equation in [6]. In the presence of a polynomial potential of degree at most 2, the time-stepping equations proposed in this work are at most quadratic also during the collision steps.…”

Continuous Finite Element Methods of Molecular Dynamics Simulations

“…The total energy , obtained by summing the kinetic and potential contributions, is a conserved quantity for the molecular system. Many numerical methods have been developed but the most effective for use in MD simulations should have superior long-term stability properties and energy conservation and permit a large integration time step [1][2][3][4][5][6]. It is important to construct discrete algorithms which preserve these basic properties.…”