Introduction to Mathematical and Computational Modelling in Natural Sciences

Lecture is dedicated to students of natural sciences (physics, chemistry, biology and interdisciplinary studies) as well as students of mathematics and informatics. Students are expected to know basics of theoretical physics and programming in C++ and/or FORTRAN.

Plan of the lecture:

1. Theory and experiment. Modelling and simulations of physical, chemical and biological processes. Reductionism.
2. Contemporary computer architectures. Superscalar work stations. High performance computers with scaleable parallel and vector-parallel architectures.
3. Trajectories in the phase-space. Classification of dynamical systems. Classical stochastic and quantum processes.
4. Born-Oppenheimer approximation. Overview of popular quantum methods: Molecular Orbital, Valence Bond and Density Functional methods.
5. Random-number generators. Monte-Carlo algorithms.
6. Molecular dynamics (MD) algorithms. Stability of the MD algorithms.
7. Simulation of discrete systems in equilibrium states:
   • Basic statistical ensembles and thermodynamic properties.
   • Microscopic picture of the pressure, temperature and heat capacity.
   • Simulations of the free energy. Thermodynamic integration.
8. Simulations of evolution of discrete systems:
   • Time-dependent correlation functions. Transport coefficients. Simple diffusion processes.
   • Viscosity and other macroscopic properties. Spatial-temporal correlation functions.
9. Overview of selected applications. Simple materials and biomolecular systems.

Literature:

1. M.P.Allen & D.J.Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford, 1989.
2. J.M.Haile, Molecular Dynamics Simulation. Elementary Methods, John Wiley & Sons, Inc., New York, 1992.
3. R.W.Hockney & J.W.Eastwood, Computer Simulation Using Particles, McGraw Hill, New York, 1981.
4. A.R.Leach, Molecular Modelling: Principles and Applications (2nd Edition), Prentice Hall; ISBN: 0582382106, 2001.
5. B. Lesyng, Simulations of Biomolecular Systems and Processes: Perspectives and Limitations, in "Modelling and Simulation: A Tool for the Next Millenium", 13th European Simulation Multiconference, June 1-4, 1999, Warsaw, Poland. A Publication of the Society for Computer Simulation International, vol. 1, pp. 26-32, 1999.