MONTE CARLO SIMULATIONS IN STATISTICAL PHYSICS

Wolfhard JANKE (Personal webpage)

Lectures:

1. Introduction to Monte Carlo simulations
2. Improved algorithms and generalised ensembles
3. 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