The aim of this lecture series is to give an overview on current state-of-the-art Monte Carlo computer simulations in classical statistical physics. After a brief summary of the main properties of phase transitions, first a few currently studied complex systems (e.g., disordered ferromagnets, spin glasses, protein folding) will be discussed that motivate the need for rather sophisticated numerical methods and illustrate some of the questions to be answered by them. All basic techniques, however, can be explained for comparatively simple Ising and Potts models, being paradigms for systems exhibiting first- and second-order phase transitions. In the first lecture, importance sampling Monte Carlo schemes using standard local update rules such as the Metropolis, heat-bath and Glauber algorithm will be briefly introduced. This provides the basis for discussing error analyses of simulation data and explaining the critical slowing down at a second-order phase transition. The different physical origin for the even more severe slowing at a first-order phase transition will also be pointed out. Next, as an important tool for finite-size scaling analyses, histogram reweighting techniques will be introduced. In the second lecture, cluster-update algorithms will be considered, which for some models can reduce critical slowing down dramatically. The geometrical interpretation of the emerging clusters leads naturally to so-called improved estimators in terms of percolation observables. Also a brief overview of other improved methods will be given (multigrid techniques, tempering algorithms, multicanonical ensemble). Finally a sample study for spin models will be presented which shows the typical procedure followed also in the more complex cases. Focussing mainly on the basic concepts, the lecture series is addressed to a broad audience of students, whose main focus may range from applied to theoretical physics. Small exercises and assignments will be assigned, that should be worked out by the students in the Lab course. To speed up the development of basic tools, basic parts of program codes and a few subroutines will be provided.

Lecture I - Introduction to Monte Carlo simulations

This lecture first motivates the scope of computer simulations in statistical physics and then introduces the basic concepts underlying Monte Carlo simulations and their statistical analysis. The power of the method can be already illustrated with Ising and Potts models.

Statistical physics and basic properties of phase transitions
Overview of currently studied complex systems (disordered ferromagnets, spin glasses, protein folding)
Importance sampling Monte Carlo schemes
Local update procedures (Metropolis, heat-bath, Glauber)
Statistical error analyses and critical slowing down
Histogram reweighting techniques

Lecture II - Cluster algorithms and applications

For certain classes of models the simulations can be dramatically improved by using more advanced algorithms. This will be illustrated with cluster-update algorithms, whose geometrical interpretation in terms of percolation observables leads to further improvements. Finally, the discussed improved methods will be applied to a simple case study, illustrating the general procedure of any Monte Carlo simulation study.

Cluster algorithms (Swenden-Wang and Wolff)
Percolation interpretation and improved estimators
Geometric vs. stochastic clusters
Brief overview of other improved methods(multigrid methods, tempering algorithms, multicanonical ensemble)
Applications to a simple example system

Recent textbooks on the subject include:

B.A. Berg,
Markov Chain Monte Carlo Simulations and Their Statistical Analysis,
World Scientific, Singapore, 2004.
K. Binder and D.W. Heermann,
Monte Carlo Simulations in Statistical Physics: An Introduction, 4th edition,
Springer, Berlin, 2002.
D.P. Landau and K. Binder,
Monte Carlo Simulations in Statistical Physics,
Cambridge University Press, Cambridge, 2000.
M.E.J. Newman and G.T. Barkema,
Monte Carlo Methods in Statistical Physics,
Clarendon Press, Oxford, 1999.

Review articles covering the material of the lectures:

W. Janke,
Monte Carlo Simulations of Spin Systems,
invited lecture at the HERAEUS-School ``Physik mit dem Computer'', Chemnitz, September 18 - 29, 1995, in: Computational Physics: Selected Methods - Simple Exercises - Serious Applications, ed. K.H. Hoffmann and M. Schreiber (Springer, Berlin, 1996), p. 10 [ online ].
W. Janke,
Nonlocal Monte Carlo Algorithms for Statistical Physics Applications,
Mathematics and Computers in Simulations 47, 329 (1998); invited review lecture at the IMACS Workshop on Monte Carlo Methods , Brussels, April 1 - 3, 1997 [ online ].
W. Janke,
Pseudo Random Numbers: Generation and Quality Checks,
invited lecture notes, in: Proceedings of the Euro Winter School Quantum Simulations of Complex Many-Body Systems: From Theory to Algorithms, edited by J. Grotendorst, D. Marx, and A. Muramatsu, John von Neumann Institute for Computing, Jülich, NIC Series, Vol. 10, pp. 447-458 (2002) [ online ].
W. Janke,
Statistical Analysis of Simulations: Data Correlations and Error Estimation,
invited lecture notes, in: Proceedings of the Euro Winter School Quantum Simulations of Complex Many-Body Systems: From Theory to Algorithms, edited by J. Grotendorst, D. Marx, and A. Muramatsu, John von Neumann Institute for Computing, Jülich, NIC Series, Vol. 10, pp. 423-445 (2002) [ online ].
W. Janke,
Multicanonical Monte Carlo Simulations,
Physica A254 , 164 (1998); invited talk, reprinted in: Proceedings of StatPhys-Tapei-1997 New Directions in Statistical Physics , Academia Sinica, Taipei, Taiwan, August 3 - 11, 1997, edited by C.-K. Hu and K.-t Leung (Elsevier Science, Amsterdam, 1998); p. 164 [ online ].
W. Janke,
Histograms and All That,
in: Computer Simulations of Surfaces and Interfaces, NATO Science Series, II. Mathematics, Physics and Chemistry - Vol. 114, Proceedings of the NATO Advanced Study Institute, Albena, Bulgaria, 9 - 20 September 2002, edited by B. Dünweg, D.P. Landau, and A.I. Milchev (Kluwer, Dordrecht, 2003), pp. 137 - 157 [ online].
W. Janke,
First-Order Phase Transitions,
in: Computer Simulations of Surfaces and Interfaces, NATO Science Series, II. Mathematics, Physics and Chemistry - Vol. 114, Proceedings of the NATO Advanced Study Institute, Albena, Bulgaria, 9 - 20 September 2002, edited by B. Dünweg, D.P. Landau, and A.I. Milchev (Kluwer, Dordrecht, 2003), pp. 111 - 135 [ online].
W. Janke, P.-E. Berche, C. Chatelain, and B. Berche,
Phase Transitions in Disordered Ferromagnets,
in: Proceedings of NIC-Symposium 2004, Proceedings, edited by D. Wolf, G. Münster, and M. Kremer, John von Neumann Institute for Computing, Jülich, NIC Series, Vol. 20, pp. 241 - 250 (2003) [ online ].
W. Janke and M. Weigel,
Monte Carlo Studies of Connectivity Disorder,
in: High Performance Computing in Science and Engineering, Munich 2004, transactions of the Second Joint HLRB and KONWIHR Result and Reviewing Workshop, March 2nd and 3rd, 2004, Technical University of Munich (Springer-Verlag, Berlin, Heidelberg, New York, 2004), p. 363 - 373 [ online].
W. Janke, B.A. Berg, and A. Billoire,
Multi-Overlap Simulations of Spin Glasses,
in: NIC Symposium 2001, edited by H. Rollnik and D. Wolf, John von Neumann Institute for Computing, Jülich, NIC Series Vol. 9, pp. 301-314 (2002) [ online ] [ cond-mat/0112036].
M. Bachmann and W. Janke,
Thermodynamics of Protein Folding from Coarse-Grained Models' Perspectives,
to appear in the CECAM-Workshop lecture notes Rugged Free Energy Landscapes: Common Computational Approaches in Spin Glasses, Structural Glasses and Biological Macromolecules, edited by W. Janke, Lecture Notes in Physics (Springer, Berlin, 2006) (in print).

Last Update: Mon Mar 13 23:44:53 CET 2006

Lectures at the Heraeus Summerschool Computational Many Particle Physics

Greifswald, September 18-29, 2006

Monte Carlo Methods in Classical Statistical Physics

Transparencies:

Tutorial/Assigments:

Material:

Topics:

Contents:

Lecture I - Introduction to Monte Carlo simulations

Lecture II - Cluster algorithms and applications

Recent textbooks on the subject include:

Review articles covering the material of the lectures:

Lectures at the Heraeus Summerschool
Computational Many Particle Physics