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Fitting stochastic differential equations to molecular dynamics data
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Pokern, Yvo (2007) Fitting stochastic differential equations to molecular dynamics data. PhD thesis, University of Warwick.
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WRAP_Theses_Pokern_2007.pdf - Submitted Version - Requires a PDF viewer. Download (5Mb) | Preview |
Official URL: http://webcat.warwick.ac.uk/record=b2116180
Abstract
The thesis consists of three main parts. Firstly, a molecular dynamics and potential energy minimisation package that has been implemented is described in detail All potential and force interactions are described and tested successfully. Compound tests on minimal energies for clusters of water molecules, the radial distribution function for liquid argon and the equilibrium distribution for the dihedral angle in Butane under Langevin dynamics are performed and the presence of multiple time scales is noted for Butane as well as for a simplified protein model due to Grubmiiller and Tavan.
Secondly, fitting stochastic differential equations (SDEs) to time series is studied. Initially, I consider the well-understood case of non-degenerate diffusions, where all components of the process are driven directly by Brownian motion. An SDE with constant diffusivity and trigonometric force expression is fitted to trajectories obtained from simulations of Butane by maximum likelihood methods and fitted diffusion and drift parameters depend strongly on the timescale considered. Hypoelliptic diffusion processes are considered next. Here, the unexpected failure of simple estimators necessitates the use of carefully chosen approximate likelihoods. For the case of only partial observations being available, a compound algorithm is designed and numerically seen to be asymptotically consistent. It is applied to the same Butane sample path and found to equilibrate, although the fitted SDE fails to reproduce the free energy landscape
Thirdly, connections between maximum likelihood estimators (MLEs) and practitioners’ methods are investigated. Analytical links are found for reversible processes and for second order Langevin processes. In the case of ID processes, MLE and practitioners’ methods for the drift are found to yield estimators identical up to lower order terms even for finite times of observation.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics Q Science > QD Chemistry |
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Library of Congress Subject Headings (LCSH): | Molecular dynamics -- Mathematics, Stochastic differential equations, Brownian motion processes -- Mathematical models | ||||
Official Date: | February 2007 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Stuart, Andrew (Professor of mathematics) | ||||
Format of File: | |||||
Extent: | xi, 141 leaves : illustrations | ||||
Language: | eng |
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