MONTE CARLO SIMULATIONS IN STATISTICAL PHYSICS
Wolfhard JANKE
Institut für Theoretische Physik, Universität Leipzig
Lectures:
- Introduction to Monte Carlo simulations
- Improved algorithms and generalised ensembles
- Applications to disordered systems
Contents: The aim of this lecture series is to give an overview on the current state-of-the-art of Monte Carlo computer simulations and to illustrate them in the first two lectures with simple applications to the Ising model of statistical physics. After reviewing in the first lecture importance sampling Monte Carlo schemes based on Markov chains and standard local update rules such as the Metropolis and heat-bath algorithm, statistical error analyses of simulation data and critical slowing down at a second-order phase transition will be discussed. As an important tool for finite-size scaling analyses, histogram reweighting techniques are introduced.
Next advanced update algorithms will be considered which, for certain classes of models, can drastically improve the performance of simulations. This will be illustrated with cluster-update algorithms, reducing critical slowing down at second-order phase transitions, and multicanonical simulations, greatly improving simulations at first-order phase transitions and, in general, for systems with rare-event states. A few other useful methods will be briefly mentioned.
Mainly intended as an outlook, the third lecture will be devoted to more advanced applications to disordered systems such as diluted ferromagnets, random lattices and spin glasses which in general require especially tailored algorithms for their successful simulation.
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 referring mainly to the first and partly the second lecture will be assigned, that should be worked out by the students.
Lecture I - Introduction to Monte Carlo simulations: This lecture introduces the basic concepts underlying Monte Carlo simulations and their statistical analysis. The power of the method will be illustrated for the Ising model.
- Importance sampling Monte Carlo
- Local update procedures (Metropolis, heat-bath)
- Statistical error analyses (critical slowing down)
- Histogram reweighting techniques
- Applications to the Ising model
Lecture II - Improved algorithms and generalised ensembles: For certain classes of models the simulations can be drastically improved by using more advanced algorithms. This will be illustrated with cluster-update algorithms, drastically reducing critical slowing down at second-order phase transitions, and multicanonical simulations, greatly improving simulations at first-order phase transitions and, in general, for systems with rare-event states. Other useful methods will be only briefly mentioned.
- Cluster algorithms
- Multigrid methods
- Generalized ensembles (multicanonical simulations etc.)
- Simulated and parallel tempering
Lecture III - Applications to disordered systems: Numerical simulations of quenched, disordered systems (e.g. random-bond or diluted ferromagnets, random lattices or graphs, spin glasses) in general require especially tailored algorithms in order to achieve reliable results in reasonable computing times (which are usually still large, even on supercomputers). Methodological similarities to the problem of protein folding will be sketched. The objective of this lecture is to give an outlook to computer experiments for such systems and to illustrate them by specific examples without going too much into the details.
- Diluted ferromagnets
- Random lattices or graphs
- Spin glasses
- Protein folding
Recent textbooks on the subject include:
[1] M.E.J. Newman, G.T. Barkema. Monte Carlo Methods in
Statistical Physics. Clarendon Press, Oxford, 1999.
[2] D.P. Landau, K. Binder. {Monte Carlo Simulations in
Statistical Physics. Cambridge University Press, Cambridge, 2000.
[3] K. Binder, D.W. Heermann. Monte Carlo Simulations in
Statistical Physics: An Introduction (4th edition). Springer,
Berlin, 2002.
[4] B.A. Berg. Markov Chain Monte Carlo Simulations and Their
Statistical Analysis. World Scientific, Singapore, 2004.
A few review articles covering the material of the lectures are (see also
link):
[5] W. Janke. Nonlocal Monte Carlo Algorithms for Statistical
Physics Applications. Math. Comput. Simulat. 47 (1998) 329.
[6] 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,
Julich, NIC Series, Vol. 10 (2002) 423-445.
[7] W. Janke. Multicanonical Monte Carlo Simulations. Physica A,
254 (1998) 164.
[8] 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. Donweg, D.P. Landau, A.I. Milchev. Kluwer, Dordrecht,
114 (2003) 137-157.
[9] W. Janke. First-Order Phase Transitions. In: Computer
Simulations of Surfaces and Interfaces, NATO Science Series, II.
Mathematics, Physics and Chemistry. Proceedings of the NATO Advanced
Study Institute, Albena, Bulgaria, 9 - 20 September 2002. Edited by
B. Donweg, D.P. Landau, A.I. Milchev. Kluwer, Dordrecht, 2003. -
Vol. 114, Pp. 111-135.
[10] W. Janke, et al. Phase transitions in disordered ferromagnets.
In: Proceedings NIC-Symposium 2004. Edited by D. Wolf, et al.
Institute for Computing, Julich, NIC Series, 20 (2003)
241-250.
[11] W. Janke, B.A. Berg, A. Billoire. Multi-overlap simulations of
spin glasses. In: Proceedings NIC-Symposium 2004. Edited by D.
Wolf, et al. Institute for Computing, Julich, NIC Series, 20
(2003) 301-314.
[12] W. Janke, 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, Pp. 363-373.
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