Leimkuhler, Benedict

Gyongy, Istvan

Tanner, Jared

Matthews, Charles

Engineering and Physical Sciences Research Council (EPSRC)

2014-06-11T14:45:39Z

2014-06-11T14:45:39Z

2013-11-28

Molecular dynamics (MD) computations aim to simulate materials at the atomic level by approximating molecular interactions classically, relying on the Born-Oppenheimer approximation and semi-empirical potential energy functions as an alternative to solving the difficult time-dependent Schrodinger equation. An approximate solution is obtained by discretization in time, with an appropriate algorithm used to advance the state of the system between successive timesteps. Modern MD simulations simulate complex systems with as many as a trillion individual atoms in three spatial dimensions. Many applications use MD to compute ensemble averages of molecular systems at constant temperature. Langevin dynamics approximates the effects of weakly coupling an external energy reservoir to a system of interest, by adding the stochastic Ornstein- Uhlenbeck process to the system momenta, where the resulting trajectories are ergodic with respect to the canonical (Boltzmann-Gibbs) distribution. By solving the resulting stochastic differential equations (SDEs), we can compute trajectories that sample the accessible states of a system at a constant temperature by evolving the dynamics in time. The complexity of the classical potential energy function requires the use of efficient discretization schemes to evolve the dynamics.In this thesis we provide a systematic evaluation of splitting-based methods for the integration of Langevin dynamics. We focus on the weak properties of methods for confiurational sampling in MD, given as the accuracy of averages computed via numerical discretization. Our emphasis is on the application of discretization algorithms to high performance computing (HPC) simulations of a wide variety of phenomena, where configurational sampling is the goal. Our first contribution is to give a framework for the analysis of stochastic splitting methods in the spirit of backward error analysis, which provides, in certain cases, explicit formulae required to correct the errors in observed averages. A second contribution of this thesis is the investigation of the performance of schemes in the overdamped limit of Langevin dynamics (Brownian or Smoluchowski dynamics), showing the inconsistency of some numerical schemes in this limit. A new method is given that is second-order accurate (in law) but requires only one force evaluation per timestep. Finally we compare the performance of our derived schemes against those in common use in MD codes, by comparing the observed errors introduced by each algorithm when sampling a solvated alanine dipeptide molecule, based on our implementation of the schemes in state-of-the-art molecular simulation software. One scheme is found to give exceptional results for the computed averages of functions purely of position.

http://hdl.handle.net/1842/8949

en

The University of Edinburgh

B. Leimkuhler and C. Matthews. Rational construction of stochastic numerical methods for molecular sampling. Applied Mathematics Research eXpress, 2013(1):34{56, 2013.

B. Leimkuhler and C. Matthews. Robust and e cient con gurational molecular sampling via Langevin dynamics. The Journal of Chemical Physics, 138(17):174102, 2013.

canonical sampling

Langevin dynamics

stochastic differentiable equations

splitting schemes

Error in the invariant measure of numerical discretization schemes for canonical sampling of molecular dynamics

Thesis or Dissertation

Doctoral

PhD Doctor of Philosophy