Mathematical and computational methods from micro to macro scales, 8 credits

Stochastic molecular dynamics of liquid-solid phase transition

Description

The Schrödinger equation, which accurately models nuclei-electron system without unknown parameters, is the basis for solid state physics and computational chemistry. An important issue is its high computational complexity, e.g. already for a water molecule it means to solve a partial differential equation in 39 dimensions. Computational approximations are therefore needed and the goal of the course is to understand the accuracy and numerical complexity of important coarse-grained approximations.

The complexity is reduced by classical approximation of the nuclei, using surface-hopping, Ehrenfest or the Born-Oppenheimer dynamics. To computationally solve the quantum problem for the electrons the Hartree-Fock and Kohn-Sham density functional theory is important and leads to an ab intio molecular dynamics model. The ab initio molecular dynamics can be simplified by empirical potentials. Thermal fluctuations in an ensemble at constant temperature introduces stochastics into the dynamics which leads to the Langevin molecular dynamics, or variants thereof. On long time scales and the high friction limit this dynamics can be described without the velocities by the Smoluchowski equation. The next step in the coarse-graining process is to derive partial differential equations -- for the mass, momentum and energy of a continuum fluid -- from Langevin or Smoluchowski molecular dynamics, which determines the otherwise unspecified pressure, viscosity and heat conductivity; we present an example of such a coarse-graining process in the case of modelling a solid-liquid melt.

Contents

1. The Schrödinger equation
   
   1. Introduction, postulates
   2. Properties (conservation of L2-norm, symmetries, relation with classical mechanics, etc.)
   3. Approximations
      1. Surface-hopping, Ehrenfest and Born-Oppenheimer
      2. Hartree-Fock
      3. Density functional theory (Kohn-Sham)
      4. Car-Parrinello, Langevin molecular dynamics
      5. Bridging ab initio and empirical methods
   
   

   Literature:	[AS]
   	[CL]

   
   
2. Molecular dynamics
   
   1. Thermodynamics and statistical mechanics
   2. Micro- / canonical ensemble
   3. Molecular dynamics simulation
   4. Reaction rates and reaction paths
   5. Illustration in solids and liquids: diffractograms, structure function, etc
3. Continuum problems
   
   1. A phase-field continuum model derived from Smoluchowski MD
   
   

   Literature:	[MD], [AS]

Schedule

Teachers

Examination

The projects are made in groups of two and consist in presenting a scientific article in class. The presentations should be 30 minutes (max) and include problem formulation, theoretical background, results and any other interesting points you want to make. The material should be adapted to the audicence: your classmates. Make sure the level is such that they can understand everything and learn something new. You are encouraged to redo computations in the paper and to ask the teachers for input and comments during the preparations.

Note: Slides should be used. After the presentation they will be printed and distributed to the whole class. The slides will be part of the course literature.

Please use this opportunity to practice the difficult art of holding a seminar. Remember that presenting a material in a convincing and clear way is important and requires good preparation. For this we all need practice and constructive criticism, students as well as teachers.

1. Cances, Castella, Chartier, Faou, Le Bris, Legoll and Turinici. Higher-order averaging schemes with error bounds for thermodynamical properties calculations by MD simulations, INRIA report 4875, 2003. (To appear in J. Chem. Phys. 2004.)
2. Katsoulakis, Majda and Vlachos. Coarse-grained stochastic processes and Monte Carlo simulations in lattice system. J. Comp. Phys., 186:250-278, 2003.
   Katsoulakis, Majda and Vlachos. Coarse-grained stochastic processes for microscopic lattice system. Proc. Nat. Acad. Sci., 100(3):782-787, 2003.
3. Cances, et al. Control of Molecular Systems. [CLB], 246-253.
   
4. Bandrauk and Chelkowski. Assymetric electron-nuclear dynamics in two-color laser fields; laser phase directional control of photofragments in H2. Phys. Rev. Lett. 84:3562-3565, 2000.
   LeBris et al., Quantum Control, AMS.
5. Bornemann and Schutte. A mathematical investigation of the Car-Parrinello method. Numer. Math. 78:359-376, 1998.
6. Schutte and Huisinga. Biomolecular conformations can be identified as metastable sets of molecular dynamics, [CLB], 699-745.
   
7. Huisinga, Schutte and Stuart. Extracting macroscopic stochastic dynamics: model problems. Comm. Pure. App. Math 57:234-269, 2003.
   
8. Cances E., Legoll F and Stolz G., Theoretical and numerical comparison of some sampling methods for molecular dynamics, Math. Model. Num. Anal., 41 (2007) 351-389.
   
9. [CLB] Cances E., Defranceschi M., Kutzelnigg W., LeBris C., Maday Y., Computational Chemistry: a primer, n Handbook of Numerical Analysis, X, North-Holland 2003.
10. Metzner, Ph. and Schütte, Ch. and Vanden-Eijnden, E. (2008) Transition Path Theory for Markov Jump Processes. Mult. Mod. Sim.
    
11. W. E, B. Engquist, X. Li, W. Ren and E. Vanden-Eijnden. Heterogeneous multiscale methods: A review. Comm. Comput. Phys., vol. 2, no. 3, pp. 367-450, 2007.
    
12. W. E and J.F. Lu, The continuum limit and QM-continuum approximation of quantum mechanical models of solids. Comm. Math. Sci., vol. 5, no. 3, pp. 679-696, 2007.
    
13. W. Ren and W. E. Heterogeneous multiscale method for the modeling of complex fluids and micro-fluidics. J. Comput. Phys., vol. 204, no. 1, pp. 1-26, 2005.
    
14. Finite temperature string method for the study of rare events, W. E, W. Ren, and E. Vanden-Eijnden, J. Phys. Chem. B 109, 6688 (2005)
    
15. Analytical and numerical study of coupled atomistic-continuum methods for fluids, W. Ren, J. Comput. Phys. 227, 1353 (2007)
    
16. T. LeLievre, M. Rousset and G. Stoltz, Computation of free energy profiles with parallel adaptive dynamics, Journal of Chemical Physics 126, 134111 (2007).
    
17. E. Cancès, B. Jourdain and T. Lelievre, Quantum Monte Carlo simulations of fermions. A mathematical analysis of the fixed-node approximation, Mathematical Models and Methods in Applied Sciences, 16(9), 1403-1440, (2006).
    
18. KSSOLV: a MATLAB Toolbox for Solving the Kohn-Sham Equations, Chao Yang, B. Lee, J. Meza and L. W. Wang , accepted, ACM Trans. Math. Software, 2008.
    
19. Griebel, Michael; Hamaekers, Jan Sparse grids for the Schrödinger equation. M2AN Math. Model. Numer. Anal. 41 (2007), no. 2, 215--247.
20. García-Cervera, Carlos J.; Lu, Jianfeng; E, Weinan A sub-linear scaling algorithm for computing the electronic structure of materials. Commun. Math. Sci. 5 (2007), no. 4, 999--1026.
21. Bond, Stephen D.; Leimkuhler, Benedict J. Molecular dynamics and the accuracy of numerically computed averages. Acta Numer. 16 (2007), 1--65.
22. Claude LeBris , Computational chemistry from the perspective of numerical analysis, Acta Numerica, 363-444, 2005.

Literature

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