Monte Carlo Simulations in Statistical Physics and Quantum Field Theory
Martin Hasenbusch
last update 13.7.2005
Plan of the Lecture
1. Introduction
1.1 Spin Models
1.2 Numerical Integration
2. Monte Carlo Integration, "simple sampling"
2.1 Random numbers, Random number generators
2.2 Distributions of Random Numbers
2.3 Statistical Errors
2.4 Example: Random Walk
2.5 Example: Percolation
2. "Importance sampling"
3.1 Problems with "simple sampling"
3.2 "importance sampling"
3.3 Markov chain
3.4 Metropolis Algorithm
3.5 Example: Ising model
3.6 Statistical Errors and Autocorrelation times
4.
The Phase transition of the Ising model
5. Finite Size Scaling
6. Critical Slowing Down
7. Cluster algorithm: Ising model
7.1 Identifying Clusters: Hoshen-Kopelman and
"Ants in the labyrinth".
8. The Cluster algorithm applied to other models
8.1 Landau-Ginzburg model on the lattice
8.2 O(N)-invariant non-linear sigma-models
8.3 Interface models
8.4 Limitations of the Cluster algorithm
9. Metropolis, Heat-bath and Overrelaxation applied to the
classical Heisenberg model
10. First order Transitions
11. Renormalisation group: Monte Carlo Methods
Lecture Notes
Exercises:
I
II
IV
Program to generate all configurations for the 2D Ising model on a small lattice
V
Metropolis+Swendsen-Wang-Cluster program
VI and VII
VIII
Discussion of a real-space RG-transformation for the 2D Ising model,
Fortran77 program performing the RG-transformation discussed in the
note above.
Links:
Alan Sokals lecture can be found
here
Xtoys by
M. Creutz
A java applet to demonstrate three algorithms on the Ising model
Search with google:
Monte+Carlo+Simulation