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DRPS : Course Catalogue : School of Mathematics : Mathematics

Undergraduate Course: Molecular Dynamics (MATH10084)

Course Outline
SchoolSchool of Mathematics CollegeCollege of Science and Engineering
Credit level (Normal year taken)SCQF Level 10 (Year 4 Undergraduate) AvailabilityAvailable to all students
SCQF Credits10 ECTS Credits5
SummaryMolecular dynamics is an area of high current interest throughout science and engineering, being used to simulate diverse applications (HIV drugs, DNA helices, liquid crystals, polymers, and carbon nanotubules, to name a few). The topic concerns the treatment of Hamiltonian mechanical systems, with or without constraints, and either in isolation or in contact with a stochastic thermal bath.

The course will start with Newton¿s equations for a system of interacting particles. Different potential energy functions are used to model the effects of short-ranged repulsion, dispersion, Coulombic attraction, and bonding. These will be surveyed. The generic properties of molecular N-body systems will then be considered from the perspective of dynamical systems, including equilibrium states, normal modes, variational equations, and Lyapunov exponents.

Geometric integrators based on the symplectic property of Hamiltonian flows will be constructed using splitting and these will be analyzed using the Lie series technique (Baker-Campbell-Hausdorff expansion), providing justification for the long term stability of the methods and their approximate conservation of energy. These ideas will be extended to systems with holonomic constraints.

The last part of the course will examine the stochastic differential equations used to model heat baths in molecular modeling. The equations of motion will be discussed in terms of their diffusion properties beginning with an elementary derivation of random walks, Brownian dynamics, and Fokker-Planck equations. The focus here will be on the design and implementation of numerical methods without a detailed analysis of their properties.

The course will include a numerical project using MATLAB code to simulate a small molecular system.
Course description 1. Overview of molecular potentials: (bonds, van der Waals interactions, Coulomb potentials, and certain coarse grained models)
2. Overview of basic concepts in classical mechanics/ODEs: Lagrangian and Hamiltonian systems, first integrals, equilibrium, linearization, stability, variational equations
3. Numerical methods: convergence, concepts of accuracy, volume preserving methods, symplectic integrators. Error expansion using Lie derivative.
4. Microcanonical ensemble: Liouville's equation, evolution of phase space density, mixing property.
5. Canonical ensemble: Gibbs distribution, temperature, Brownian motion, SDEs, Ito Formula, Langevin dynamics. Numerical methods for canonical sampling. Invariant measure error. Examples of Langevin splitting methods.
Entry Requirements (not applicable to Visiting Students)
Pre-requisites Students MUST have passed: Several Variable Calculus and Differential Equations (MATH08063) OR ( Mathematics for Science and Engineering 2a (MATH08069) AND Mathematics for Science and Engineering 2b (MATH08070))
Prohibited Combinations Other requirements It is recommended that students have attended MATH10060 Numerical Ordinary Differential Equations and Applications, MATH10066 Honours Differential Equations, and MATH10067 Honours Complex Variables.
Information for Visiting Students
Course Delivery Information
Not being delivered
Learning Outcomes
Students will be able to
(1) describe the features of the most common potentials in classical molecular models, including those used in materials and biological modelling (although they probably would not be able to derive them),
(2) be able to demonstrate first integrals and the symplectic property for some classes of numerical methods,
(3) be able to describe numerical methods for constrained dynamical systems and their implementation,
(4) be able to use software to explore the chaotic dynamics of molecular models,
(5) understand and be able to use and to design splitting methods for Langevin dynamics for applications in molecular sampling.
Reading List
Recommended :
B. Leimkuhler and C. Matthews, Molecular Dynamics, Springer, to appear. [a prepublication draft will be provided if the book is not available at run-time]
M. Allen and D. Tildesley, Computer Simulation of Liquids, Clarendon Press, 1989
D. Frenkel and B. Smit, Understanding Molecular Simulation, 2nd Edition, Academic Press, 2002.
B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics, Cambridge
University Press, 2005.
Additional Information
Graduate Attributes and Skills Not entered
Course organiserDr Martin Dindos
Course secretaryMrs Alison Fairgrieve
Tel: (0131 6)50 5045
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