METHODS OF COMPUTATIONAL PHYSICS

METHODS OF COMPUTATIONAL PHYSICS (Spring 2008)

Course Number: Phys 516
Class Number: 50614R
Instructor: Aiichiro Nakano; office: VHE 610; phone: (213) 821-2657; email: anakano@usc.edu
Lecture: 10:00-10:50 M W F, GFS 216
Office Hour: 15:00-16:50 F
Prerequisites: Basic knowledge of calculus and undergraduate physics; familiarity with a programming language such as C or Fortran.
Textbooks:
T. Pang, "An Introduction to Computational Physics, 2nd Ed." (Cambridge Univ. Press, 2006) --sample C, Fortran 77, and Fortran 90 programs available on line.
W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, "Numerical Recipes, 3rd Ed." (Cambridge Univ. Press, 2007)--available online (C, Fortran 77, and Fortran 90)

Course Description
Students will learn basic elements of computational methods and acquire hands-on experience in their practical use in the context of computer simulations to solve physics problems.

Molecular dynamics simulation of the oxidation of an aluminum nanoparticle.

Syllabus

Monte Carlo (MC) simulation of spins--Ising model
- Numerical vs. MC integration: Simpson's rule, Gaussian quadrature (orthogonal functions--recursive function evaluation, generating function)
- Probability: Importance sampling, Markov chain, Metropolis algorithm
- Random number generation (RNG)
- Statistics: Variance, standard deviation, standard deviation of the MC mean
- Cluster analysis: Graphs, search, stack
MC simulation of stock price--geometric Brownian motion
- Random walk: Einstein's law, central-limit theorem
- Random variable: Black-Scholes analysis
- Coordinate transformation: Jacobian, Box-Muller algorithm for RNG of normal distribution
- Interpolation: Least square fit of data
Molecular dynamics (MD) simulation of particles--Newton's second law of motion
- Numerical differentiation
- Ordinary differential equation (ODE): Symplectic integrators
- Minimization of functions: Conjugate gradient method
- Hybrid MD/MC simulation
Quantum dynamics simulation of an electron--time-dependent Schrodinger equation
- Partial differential equation (PDE)
- Fourier analysis: Spectral analysis, fast Fourier transform (FFT)
Electronic structures of molecules--quantum mechanical eigenvalue problem
- Linear algebra: Matrix, orthogonal transformation, rank, singular value decomposition
- Matrix eigensystems: Housholder transformation, QL decomposition
- Root finding: Newton-Raphson method

Announcements

1/14/08 (M): Please email your response to the questionnaire (1. Name, 2. PhD or MS, 3. Major, 4. Your Research Topics, 5. What Computational Methods in Particular Would You Like to Learn?) from your USC account to anakano@usc.edu with subject phys516 by the end of Sunday, Jan. 20.
2/20/08 (W): An extra lecture on the Lanczos method for solving eigenvalue problems at 3:30 pm in KAP 165.
3/5/08 (W): You are welcome to participate in a tour to the USC High Performance Computing Center, starting from KAP 165 at 3:30 pm.
4/18/08 (F): Guest lecture by Prof. Elizabeth Cochran on The Quake Catcher Network: Adventures in Volunteer Computing.

Lecture Notes and References

Introduction: notes and slides
Monte Carlo basics
Monte Carlo simulation of spins
Advanced Monte Carlo simulation techniques
Numerical integration and Gaussian quadratures; Numerical Recipes, Sec. 4.5
Molecular dynamics basics
Quantum dynamics basics: splitting-operator and spectral methods; Fourier transform (Numerical Recipes, Sec. 12.1) and fast Fourier transform (Numerical Recipes, Sec. 12.2)
Monte Carlo simulation of stochastic processes
Tight binding model of electronic structures
Eigensystems
Newton method for root finding
Singular value decomposition and density matrix
Quantum Monte Carlo simulation
Kinetic Monte Carlo simulation and transition state theory
Miscellaneous lectures:

Reading List

Monte Carlo simulations: U. Wolff, Phys. Rev. Lett. 62, 361 (1989); D. Kandel, et al., Phys. Rev. Lett. 60, 1591 (1988); B. A. Berg & T. Neuhaus, Phys. Rev. Lett. 68, 9 (1992); F. Wang & D. P. Landau, Phys. Rev. Lett. 86, 2050 (2001); B. Mehlig, et al., Phys. Rev. B 45, 679 (1992)
Molecular dynamics integrator: M. Tuckerman, B. J. Berne, & G. J. Martyna, J. Chem. Phys. 97, 1990 (1992)
Higher-order symplectic integrators: H. Yoshida, Phys. Lett. A 150, 262 (1990)
Backward error analysis for numerical integrators: S. Reich, SIAM J. Numer. Anal. 36, 1549 (1999)
Transferable tight-binding models for silicon: I. Kwon, et al., Phys. Rev. B, 49, 7242 (1994)
Density-matrix algorithms for quantum renormalization groups: S. R. White, Phys. Rev. B, 48, 10345 (1993)
An introduction to Monte Carlo simulations of surface reactions: A. P. J. Jansen, arXiv:cond-mat/0303028
Theoretical foundations of dynamical Monte Carlo simulations: K. A. Fichthorn and W. H. Weinberg, J. Chem Phys. 95, 1090 (1991)
Molecular kinetics simulation: A. Nakano, Comput. Phys. Commun. 176, 292 (2007); ibid. 178, 280 (2008)

Assignments

1: Monte Carlo basics (due Friday, Feb. 1)
2: Monte Carlo simulation of spins (due Friday, Feb. 15)
3: Symplectic molecular dynamics algorithm (due Wednesday, Feb. 27)
4: Molecular dynamics (due Wednesday, Mar. 5)
5: Quantum dynamics basics (due Wednesday, Mar. 12)
6: Quantum dynamics programming--spectral method (due Friday, Mar. 28)
7: Stochastic simulation of stock price (due Friday, Apr. 4)
8: Tight binding model of electronic structures (due Monday, Apr. 14)
Final project (due Wednesday, May 14); a brief presentation on your project is required on Friday, May 2.

Source Codes

Monte Carlo basics: hit.c, mean.c, gauleg-driver.c, gauleg.c
Molecular dynamics basics: md.c, md.h, md.in
Quantum dynamics basics: qd.c, qd.h, qd.in, qd1.c, qd1.h, qd1.in, four1.c
Stochastic simulation: diffuse.c
Tight binding model of electronic structures: tb_util.c, eigen.c, singular.c, svdcmp.c