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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

Lecture on Monte Carlo Simulations by Hans Gerd Evertz

Monte Carlo Simulation for Statistical Physics by Paul Coddington.

Monte-Carlo Course Materials by Jonathan Goodman.

Xtoys by
M. Creutz

A java applet to demonstrate three algorithms on the Ising model

THE WWW VIRTUAL LIBRARY: RANDOM NUMBERS and MONTE CARLO METHODS

Search with google:

Monte+Carlo+Simulation

Programming in C

Scientific programming in C
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