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Computersimulationen in der Statistischen Physik

Vorlesung im Wintersemester 2007/2008
N. Blümer, P. Virnau

Computer simulations in statistical physics

Lecture in winter semester 2007/2008
N. Blümer, P. Virnau

Stundenzahl: 3 V + 1 Ü

Zeit und Ort: Di 10 - 12 Uhr, Lorentzraum (Staudingerweg 7, 05-127), 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 2007/2008

Sprechstunde: jeweils nach der Vorlesung und nach Vereinbarung Lecture hours: 3 V + 1 Ü (lectures + tutorials)

Time and Place: Tuesdays 10 - 12 am, Lorentzraum (Staudingerweg 7, 05-127), 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 2007/2008

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 0 (2007/10/23)
- Overview over contents of lecture
- Questionnaire
- Introductory beamer presentations
  - Overview and research areas by P. Virnau
  - Quantum Monte Carlo simulations of strongly correlated electron systems within dynamical mean-field theory by N. Blümer
Lecture 1 (2007/10/25) comp-sim-ws0708-v01.pdf
- Start Chapter I: Statistical analysis of time series
- Trace, transient, autocorrelation, probability distribution, moments, probability distributions of sums/averages, central limit theorem
- Unbiased estimators for mean and variance for data set without autocorrelation
Lecture 2 (2007/10/30) comp-sim-ws0708-v02.pdf
- Improved estimator for variance for data set with autocorrelation
- Estimator for variance / standard deviation of average
- Practical data analysis using the programs gnuplot and stats
Lecture 3 (2007/11/6+8) comp-sim-ws0708-v03.pdf
- Solutions for HW (statistical data analysis): comp-sim_hw2a.pdf
- Random number generators
Lecture 4 (2007/11/8) awk_tutorial.pdf
- Tutorial: line-oriented computer language AWK
- Linux help: Befehlsreferenz Uni Mnster
Lecture 5 (2007/11/13) awk_tutorial.pdf
- Continuation AWK tutorial
- Exercises: RNG + tests, Monte Carlo with AWK
Lecture 6 (2007/11/15)
- Solutions to exercises
- Introduction to Monte Carlo simulations: MC_Metropolis.pdf
Lecture 7 (2007/11/20)
- Introduction to Monte Carlo simulations, continued: MC_Metropolis.pdf, MC_Einfuehrung_deutsch.pdf, handout_20.11.pdf, Exercises_MC_Ising.pdf
Lecture 8 (2007/11/22) comp-sim-ws0708-v08.pdf
- Ising model: history, hamiltonian, interpretation, and relation to more general spin models
- Excursion: statistical physics in the canonical ensemble
- Phase transitions, thermodynamic limit, and critical exponents
- Boundary conditions for finite systems
- Metropolis Monte Carlo for the Ising model
Lecture 9 (2007/11/27) comp-sim-ws0708-v09.pdf
- Addendum: impossibility for importance-sampline MC to measure Z, F, or S
- Ising model: mean-field solution, solutions in d=1 and d=2
- Ising model: critical temperatures and critical exponents
- Finite-size scaling (FSS)
Lecture 10 (2007/11/29) comp-sim-ws0708-v10.pdf
- Finite-size scaling, continued; Binder cumulant
- Solution of home work: 2d Ising model comp-sim_hw4.pdf
Lecture 11 (2007/12/04)
- Introduction to Molecular Dynamics, MD_Intro.pdf
Lecture 12 (2007/12/06)
- Exercises MD: Exercises_MD.pdf, solutions_to_exercises.pdf
- MD in NVT ensemble: MD_NVT.pdf
Lecture 13 (2007/12/11) num-meth-ss07-v07.pdf
- Cluster Monte Carlo methods: Kasteleyn-Fortuin mapping, Swendson-Wang update, Wolff update
Tutorial (2007/12/13)
- Q/A for homework (cluster MC Ising)
Lecture 14 (2007/12/18)
- solutions to homework (cluster MC Ising): comp-sim_Wolff_p.pdf
- Grand canonical MC: grandcanonical_MC.pdf
Lecture 15 (2007/12/20)
Lecture 16 (2008/01/08)
Lecture 17 (2008/01/10)
Lecture 18 (2008/01/15) comp-sim-ws0708-v18.pdf
- Solutions for homework (PV)
- Chapter ?: Quantum Monte Carlo Simulations
- Classification of QMC methods
- Path integral Monte Carlo (PIMC)
Lecture 19 (2008/01/17) comp-sim-ws0708-v19.pdf
- Path integral Monte Carlo (PIMC), continued
Lecture 20 (2008/01/22) comp-sim-ws0708-v19.pdf
- Path integral Monte Carlo (PIMC), continued
Lecture 21 (2008/01/24) comp-sim-ws0708-v21.pdf
- PIMC: Observables
- Quantum Heisenberg model
- World line quantum Monte Carlo simulations
(2008/01/29) comp-sim-ws0708-v22.pdf
- WL-QMC: plaquette weights, Metropolis/heat bath rules, energy and other observables
(2008/01/31) comp-sim-ws0708-v23.pdf
- WL-QMC: loop updates
(2008/02/05)
- VMD
Project presentations (2008/02/07)
- Seminarvortrag: Umbrella-Sampling im NPT-Ensemble (Dorothea Wilms)
- Seminarvortrag: Untersuchungen zur Dynamik von Knoten in langkettigen Polymeren (Andreas Pfister)
- Seminarvortrag: QMC Weltlinien-Algorithmus mit lokalen Updates (Benjamin Niepelt)

Problem sets / Tutorials / Sample codes

2006/10/26 Variance estimator: bias for autocorrelated data?
2007/10/30 Data analysis: analyse the following sample data (solutions: comp-sim_hw2a.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)
- set 7 (10, 100, 1000, 10000 numbers)
- set 8 (10, 100, 1000, 10000 numbers)
- set 9 (10, 100, 1000, 10000 numbers)
2007/11/22 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 20-40
- Plot Binder's 4th order cumulant U₄(T)=1-⟨M⁴⟩/(3⟨M²⟩²) and determine T_c
- Optional: determine specific heat and susceptibility at selected temperatures
- raw statistics data: directory, README
2007/12/11 Cluster algorithms for Monte Carlo simulation of 2D Ising model (solution: comp-sim_hw4.pdf, code: mc_Ising_2D_v06.c)
- Implement Wolff or Swendsen-Wang cluster update algorithm
- Compare autocorrelation times for energy and/or magnetization at the critical temperature with single-spin flip algorithm
- Plot "smart observables" (for T_c determination) near T_c and show improvements by cluster algorithm
- Discuss the melting of a magnet at T>T_c; example for freezing at T<T_c: MC_Ising_freeze.png
Sample codes and tools
- command line plotting tool: gnuplot
- 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
- (Inefficient) Monte Carlo code for 2 D square Ising model (template): mc_Ising_2D_template.c; code: mc_Ising_2D_v02.c
- MC for 2D square Ising model: new template for Wolff updates: mc_Ising_2D_v04t.c, highlighted html: mc_Ising_2D_v04t.c.html

Ankndigung (aus kommentiertem Vorlesungsverzeichnis)

Inhalt der Vorlesung:

Diese Vorlesung behandelt die grundlegenden Computersimulationsmethoden fr 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. Fr 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 Lï¿½ung 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 Hï¿½fte der Vorlesung bercksichtigen wir bei der Themenauswahl gerne Vorkenntnisse und besondere Interessen der Teilnehmer. Eine aktive Mitarbeit wird bei den (flexibel stattfindenden) ï¿½ungen erwartet; auï¿½rdem sind Semesterprojekte geplant, die von den Teilnehmern jeweils einzeln oder in kleinen Gruppen durchgefhrt und am Ende in kurzen Vortrï¿½en prï¿½entiert werden.

Geforderte Vorkenntnisse: Klassische Mechanik, QM I, Statistische Thermodynamik

Literaturangaben

Announcement (from annotated 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)
Alfred V. Aho, Brian W. Kernighan, Peter J. Weinberger: The AWK Programming Language, Addison-Wesley, 1988, ISBN 0-201-07981-X.
N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller und E. Teller: Equation of State Calculations by Fast Computing Machines, Journal of Chemical Physics 21, 1087 (1953).
D. M. Ceperley: Path integrals in the theory of condensed helium, Rev. Mod. Phys. 67, 279 - 355 (1995).

Einordnung in Studien-- bzw. Prüfungsordnung: Wahlpflichtvorlesung des Diplomstudiengangs, Vorlesung des Moduls ``Physik der Flssigkeiten und Festkï¿½per'' des Masterstudiengangs Computational Sciences

Schein: Vergabe aufgrund der Teilnahme an ï¿½ungen und Semesterarbeit

Bemerkungen: Eine thematisch anschlieï¿½nde Vorlesung .shtml">Moderne numerische Methoden der Festkï¿½perphysik ist im Sommersemester 2007 geplant. ï¿½er 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 2008. Positions for diploma and doctoral students are available on related subjects.