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Stundenzahl: 3 V + 1 Ü

Zeit und Ort: Di, Do 10 - 12 Uhr, Seminarraum A (Staudingerweg 9, 01-219)

Zielgruppe: Studierende im Hauptstudium, Doktoranden

Univis-Eintrag: siehe Liste der Lehrveranstaltungen von N. Blümer im WS 2006/2007

Sprechstunde: jeweils nach der Vorlesung und nach Vereinbarung

Hinweise: Die Vorlesung wird in englischer Sprache abgehalten.

Contents / lecture notes

• Lecture 1 (2006/10/24)
  • Questionnaire for participants: comp-sim-ws0607-fragebogen.pdf
  • Outline of lecture, general discussion of contents
  • Informal introduction to chapter 1: Statistical properties of time series (NB, lecture notes (in german): comp-sim-ws0607-v01.pdf)
• Lecture 2 (2006/10/26)
  • Continuation of chapter 1: mathematical excursion, estimates for mean and variance of data, autocorrelation (NB, lecture notes: comp-sim-ws0607-v02.pdf)
  • Unbiased variance estimator for uncorrelated data
  • Homework: check bias of variance estimator for autocorrelated data
• Lecture 3 (2006/10/31)
  • Continuation of chapter 1: variance of data and error of mean (NB, lecture notes: comp-sim-ws0607-v03.pdf)
  • Brief introduction to sample codes, sample data analysis
  • Statistics formulas
  • Homework: analysis of 6 data sets
  • Random number generators I (WP)
• Lecture 4 (2006/11/02)
  • Random number generators II (WP)
  • Analysis of RNGs in physics simulations
    • Analysis of random number generators using Monte Carlo simulation, P. D. Coddington, IJMPC 5, 547 (1994) [cond-mat/9309017]
    • Physical Tests for Random Numbers in Simulations, I. Vattulainen, T. Ala-Nissila, and K. Kankaala, Phys. Rev. Lett. 73, 2513 (1994)
• Lecture 5 (2006/11/07)
  • Solutions for homeworks: comp-sim_hw2.pdf
  • Alternative analysis method for autocorrelated data: blocking analysis
  • chapter 2: Monte Carlo simulations - simple Monte Carlo, random walks (WP)
• Lecture 6 (2006/11/09)
  • nonreversal random walks (NRRW), self-avoiding random walks (SAW), biased sampling(WP)
  • Boltzmann weight in statistical physics
  • stochastic processes
• Lecture 7 (2006/11/14)
  • Markov processes(WP)
  • Metropolis transition rule
• Lecture 8 (2006/11/16)
  • Solutions for homeworks: comp-sim_hw3.pdf
  • Ising model: history, hamiltonian, limits, visualization (NB, lecture notes: comp-sim-ws0607-v08.pdf)
• Lecture 9 (2006/11/21)
  • phase transitions: classification, impossibility in finite systems, critical exponents
  • Ising model: relation to general spin models, ..., mean-field solution
• Lecture 10 (2006/11/23)
  • Ising model, cont'd (NB, lecture notes: comp-sim-ws0607-v10.pdf)
    • solution in 1 D for open bc's (B=0)
    • solution in 1 D for periodic bc's using transfer matrices (arbitrary B)
    • sketch of solution for 2 D square lattice using duality transformation (Kramers, Wigner, 1941); critical exponents
    • For more details on analytic solutions of the Ising model (also: high- and low-temperature expansions), see chapter 4.8 of lecture notes Statistische Thermodynamik by Prof. van Dongen.
    • boundary conditions (periodic, open, screw, antiperiodic, fixed/mean-field)
    • Metropolis Monte Carlo simulation of the Ising model using single-spin flips
    • homework: MC simulation of 2 D square lattice Ising model (see below)
• Lecture 11 (2006/11/28)
  • Ising model, cont'd (NB, lecture notes: comp-sim-ws0607-v11.pdf)
    • critical temperatures for various lattice types
    • practical aspects of Monte Carlo
    • finite-size scaling
• Lecture 12 (2006/11/30)
  • Molecular dynamics ...
• Lecture 13 (2006/12/05)
  • Solutions for homeworks 4: comp-sim_hw4.pdf
• Lecture 14 (2006/12/07)
• Lecture 15 (2006/12/19)
• Lecture 16 (2006/12/21)
• Lecture 17 (2007/01/09)
• Lecture 18 (2007/01/11)
  • Solutions for homeworks 5: comp-sim_hw5.pdf
• Lecture 19 (2007/01/16)
  • Cluster MC methods: Fortuin-Kasteleyn partition function, Swendsen-Wang cluster MC algorithm, Wolff cluster MC algorithm
  • Numerical comparison of single-spin flip versus Wolff cluster updates: comp-sim_Wolf.pdf
  • Implementation: mc_Ising_2D.c.html
• Lecture 20 (2007/01/18)
  • MD simulations for Lennard Jones systems
  • Literature:
    • B. Smit, Phase diagrams of Lennard-Jones fluids, J. Chem. Phys. 96, 8639 (1992)
• Lecture 21 (2007/01/23)
• Lecture 22 (2007/01/25)
• Lecture 23 (2007/01/30)
• Lecture 24 (2007/02/01)
  • Dissipative particle dynamics
  • Quantum Monte Carlo simulations (script: see below)
• Lecture 25 (2007/02/06) (NB, lecture notes: comp-sim-ws0607-QMC1.pdf)
  • Path Integral Monte Carlo I
• Lecture 26 (2007/02/08) Seminar talk
  • Ewald summation for charged particles (Alex Baade): EwaldSummation.pdf
• Lecture 27 (2007/02/13) (NB, lecture notes: comp-sim-ws0607-QMC2.pdf)
  • Path Integral Monte Carlo II
• Lecture 28 (2007/02/15)

Problem sets / Tutorials / Sample codes

• 2006/10/26 Variance estimator: bias for autocorrelated data? - yes!
• 2006/10/30 Data analysis: analyse the following sample data (solutions: comp-sim_hw2.pdf)
  • set 1 (10, 100, 1000, 10000 numbers)
  • set 2 (10, 100, 1000, 10000 numbers)
  • set 3 (10, 100, 1000, 10000 numbers)
  • set 4 (10, 100, 1000, 10000 numbers)
  • set 5 (10, 100, 1000, 10000 numbers)
  • set 6 (10, 100, 1000, 10000 numbers)
• 2006/11/09 Random walks on a lattice (solutions: comp-sim_hw3.pdf)
  • Generate histograms of the end-to-end distance R_e for random walks (generic or nonreversal) of length 100 on a 2 d cartesic lattice (using ~10⁵-10⁶ realizations)
  • Compute the fractions of walks that are self-avoiding after the n^th step: p_SAW(n)
• 2006/11/23 Monte Carlo simulation of 2D Ising model (solutions: comp-sim_hw4.pdf)
  • Write a Monte Carlo program for computing energy and magnetization of the 2 D square Ising model using single-spin flips (possibly using the template C program linked below).
  • Compute E(T), |M(T)| in a useful temperature range for lattices with linear sizes between about 4 and 20 (or maybe 30)
  • Plot Binder's 4th order cumulant U₄(T) and determine T_c
  • Optional: determine specific heat and susceptibility at selected temperatures
  • raw statistics data: directory, README
• 2006/12/21 Molecular dynamics simulation (due date: 2007/01/11, solution: comp-sim_hw5.pdf)
  • Write MD program for Lennard-Jones atoms (possibly using the template code linked below)
  • Check program
• 2007/01/11 Molecular dynamics simulation (due date: 2007/01/18)
  • Learn to use MD program (code below or own code) for Lennard-Jones atoms
  • Check dependencies of observables on cutoff, number of particles, and time discretization
  • Compute and plot (averaged) E(T), p(T) etc. in sensible range of temperatures (realized by different target energies) for density 0.6 and 256 particles
  • Plot selected pair correlation functions and look for qualitative differences
  • ...
• sample codes:
  • Primer for statistical analysis: stats_template.c (highlighted html), stats_template.c, stats_v1_4.c
  • Simple random number generation tool: random_nums.c (highlighted html), random_nums.c
  • Program for generating generic and nonreversive random walks; computes probabilities for self-avoiding walks (SAWs): random_walk.c
  • (Inefficient) Monte Carlo code for 2 D square Ising model (template): mc_Ising_2D_template.c, code: mc_Ising_2D_v02.c
  • Simple Molecular dynamics code for Lennard-Jones fluid (template): MD_LJ_v05a.c, code: MD_LJ.c, both use pointer_utils.c; for explanations, see comp-sim_hw5.pdf
  • Monte Carlo code for 2 D square Ising model with Metropolis single-spin flips oder Wolf cluster updates: mc_Ising_2D.c

Ankündigung (aus kommentiertem Vorlesungsverzeichnis)

Inhalt der Vorlesung:

Insbesondere in der zweiten Hälfte der Vorlesung berücksichtigen wir bei der Themenauswahl gerne Vorkenntnisse und besondere Interessen der Teilnehmer. Eine aktive Mitarbeit wird bei den (flexibel stattfindenden) Übungen erwartet; außerdem sind Semesterprojekte geplant, die von den Teilnehmern jeweils einzeln oder in kleinen Gruppen durchgeführt und am Ende in kurzen Vorträgen präsentiert werden.

Geforderte Vorkenntnisse: Klassische Mechanik, QM I, Statistische Thermodynamik

Literaturangaben

• M. P. Allen, D. J. Tildesley: Computer Simulation of Liquids, Oxford Science Publications, 1997
• D. P. Landau, K. Binder: A Guide to Monte Carlo Simulations in Statistical Physics, Cambridge University Press, 2005 (eBook, Hardback)
• D. Frenkel, B. Smit: Understanding Molecular Simulations, From Algorithms to Applications, Academic Press, San Diego, 2002
• D. Ceperley: Microscopic Simulations in Physics, RMP 71, S438 (1999)

Schein: Vergabe aufgrund der Teilnahme an Übungen und Semesterarbeit

Bemerkungen: Eine thematisch anschließende Vorlesung Moderne numerische Methoden der Festkörperphysik ist im Sommersemester 2007 geplant. Über das Gebiet werden Diplom- und Doktorarbeiten vergeben.