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Computersimulationen in der Statistischen Physik

Vorlesung im Wintersemester 2006/2007
N. Blümer, W. Paul

Computer simulations in statistical physics

Lecture in winter semester 2006/2007
N. Blümer, W. Paul

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 Lecture hours: 3 V + 1 Ü (lectures + tutorials)

Time and Place: Tuesdays, Thursdays 10 - 12 am, Seminarraum A (Staudingerweg 9, 01-219)

Target group: physics students after Vordiplom, PhD students

Univis entry: see list of lectures by N. Blümer in WS 2006/2007

Consultation hours: after lectures or by appointment

Hinweise: Die Vorlesung wird in englischer Sprache abgehalten.

Notes: The lecture is given in english.

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: .shtml">stats_template.c (highlighted html), stats_template.c, stats_v1_4.c
- Simple random number generation tool: .shtml">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:

Diese Vorlesung behandelt die grundlegenden Computersimulationsmethoden für Systeme vieler Teilchen, insbesondere in der Theorie der kondensierten Materie. Im Bereich der Probleme der klassischen Physik sind die wesentlichen beiden Simulationsmethoden die Molekulardynamik-Simulation und die Monte-Carlo-Simulation. Die erste ist eine deterministische Methode, die zweite eine stochastische. Für beide gilt es, den theoretischen Hintergrund zu verstehen, die Grundlagen ihrer numerischen Umsetzung zu erarbeiten und die Tricks und Fallen in der praktischen Anwendung kennenzulernen. Beide Methoden lassen sich zur numerischen Lösung von Pfadintegralen auf quantenmechanische Vielteilchensysteme verallgemeinern; gegen Ende der Vorlesung sollen diese fortgeschrittenen Aspekte zur Sprache kommen und insbesondere die Prinzipien der wichtigsten Quanten-Monte-Carlo-Algorithmen vorgestellt werden.

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

Announcement (from annoted list of lectures)

This lecture covers the basic computer simulation methods for systems of many particles, particularly as relevant in condensed matter theory. The focus is on molecular dynamics and Monte Carlo simulations, the essential methods for classical statistical physics. While the former is a deterministic method, the latter is stochastic. Goal of the lecture is to convey for both methods an understanding of the theoretical background and the principles of numerical implementations; however, we also want to discuss the most relevant tricks and possible pitfalls. Both methods can be generalized for the solution of path integrals of quantum many-particle systems; these advanced topics are planned to be addressed in the later parts of the lecture. In particular, we will present the principles of the most important quantum Monte Carlo algorithms.

We will gladly adjust the selection and depth of topics depending on prior knowledge and interests of the participants, in particular in the second half of the semester. Active participation in the tutorials is expected. In addition, the students will be asked to carry out semester projects (by themselves or in small groups) and to present the results in short talks.

Prerequisites: Classical mechanics, quantum mechanics I, statistical thermodynamics

Literature

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)

Einordnung in Studien-- bzw. Prüfungsordnung: Wahlpflichtvorlesung des Diplomstudiengangs, Vorlesung des Moduls ``Physik der Flüssigkeiten und Festkörper'' des Masterstudiengangs Computational Sciences

Schein: Vergabe aufgrund der Teilnahme an Übungen und Semesterarbeit

Bemerkungen: Eine thematisch anschließende Vorlesung .shtml">Moderne numerische Methoden der Festkörperphysik ist im Sommersemester 2007 geplant. Über das Gebiet werden Diplom- und Doktorarbeiten vergeben. Classification according to university regulations (in German): Wahlpflichtvorlesung des Diplomstudiengangs, Vorlesung des Moduls ``Physik der Flüssigkeiten und Festkörper'' des Masterstudiengangs Computational Sciences

Credits: Awarded for participation in tutorials and for semester project

Comments: A subsequent lecture on .shtml">Modern numerical methods of solid state physics is planned for the summer semester 2007. Positions for diploma and doctoral students are available on related subjects.