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Oral Qualifying Exam on Computational/Simulation Methods in Materials Science
Non-exhaustive Topical Areas in Computational/Simulation Methods:
- Molecular Dynamics: integration algorithms (including error estimation), static and dynamic time and space correlation functions and their connection to order and transport (e.g., one-body, two-body, etc., correlations, g(r), S(K), auto-velocity correlation functions).
- Advanced Topics (if appropriate): Classical MD techniqies for quantum-based methods, e.g. Car-Parinello and Born-Oppenheimer surfaces.
- Monte Carlo and Random Walks: Metropolis Algorithm, variance reduction, Markov chains.
- Random Number Generation, Sampling from an Arbitrary Probability Distributions (e.g., uniform, Poisson, Gaussian, etc.), and Histogram Re-weighting Methods.
- Ising model: Properties, Transitions, and handling boundary conditions (roughening, interfaces, etc.)
- Advanced Topics (if appropriate): Kinetic Monte Carlo, heat diffusion, Brownian motion, etc.
- Error analysis: statistical and systematic errors, bias, minimization of errors, variance reduction, etc., for Monte Carlo and Molecular Dynamics methods.
- Canonical, micro-canonical, grand canonical ensembles: constants of motion. How to address these computationally.
- Phase Transitions: solidification (phase changes), interfacial diffusion, adsorption-desorption, percolation, defects, etc.
- Techniques for long-range interactions: Coulomb sums, etc.
- General Techniques: Rudiments of Finite Difference, Finite Element
- Advanced Topics (as appropriate): Laplace transforms, or Fourier transforms for solution of exact or phenomenological equations, e.g. heat transfer, diffusion, Navier-Stokes.
- Rudimentary numerical techniques: numerical analysis, numerical integration techniques (Chebyshev or Gaussian, etc.), .
- Other Topics (if appropriate): predictor-corrector algorithms, Newton-Raphson (roots), and Gauss-Seibl (system of equations), for example.
- Advanced Topics (if appropriate): Optimization Techniques, such as Genetic Algorithms and simulated annealing.
Level (based upon UIUC courses)
MSE 485: Atomic-Scale Simulations
See "Numerical Recipes" reference below, for it contains most algorithmic components that you may have missed in your educational background.
Standard Texts
Simulation Methods/Theoretical Underpinnings
Allen and Tildesley, "Computer Simulation of Liquids" (Oxford)
Frenkel and Smit, "Understanding Molecular Simulation" (Academic Press)
Gould and Tobochnik, "Computer Simulation Methods", vol. 1 and 2 (Addison-Wesley)
J.M. Haile, "Molecular Dynamics Simulation" (Wiley)
Kalos and Whitlock, "Monte Carlo Methods" vol. 1 (Wiley)
D. Chandler, "Introduction to Modern Statistical Mechanics" (Oxford)
Numerical Techniques
Schaum's Outline on "Numerical Analysis"
D. Knuth, "The Art of Computer Programming" vol. II (Addison-Wesley)
Press, Flannery et al., "Numerical Recipes: the Art of Scientific Computing."