# Hard disks: A Window into the World of Stat Physics

### From Werner KRAUTH

This is the homepage for the Lecture series on **Hard disks: A Window into the World of Statistical Physics** given in January and February 2017 at the University of Tokyo as an intensive graduate lecture course in the School of Sciences.

Look here for practical information

## Course Title

Hard Disks: A Window into the World of Statistical Physics

## Course Objectives/Overview

The hard-disk model has exerted outstanding influence on statistical physics, starting with the first ever N-body calculation in science, by D. Bernoulli (1738), and leading up to its role in the very definition of statistical mechanics, by Maxwell and Boltzmann, and to the formal proof of the validity of statistical mechanics, by Sinai (1970). Decades ago, hard disks were the first system to be studied by Markov-chain Monte Carlo methods (Metropolis et al, 1953) and by molecular dynamics (Alder and Wainwright, 1957). Up to the present day, the model has continued to drive the development of algorithms (Bernard et al, 2009), including the "Beyond-Metropolis" approach (Michel et al, 2014). It was in hard disks, through numerical simulations, that a two-dimensional melting transition was first seen to occur (Alder and Wainwright, 1962) even though such systems cannot develop long-range crystalline order (Mermin and Wagner, 1966). This provided the starting point for the Kosterlitz-Thouless theory (1973), although the hard-disk phase diagram was established only recently (Bernard and Krauth, 2011), thus providing a clear view on the physics of phase transitions in two dimensions.

The objective of this course will thus be to discuss the aforementioned central topics of statistical physics from the unique vantage point of the hard-disk model, and in a self-contained way that will be accessible to a wide audience. Overall, the course will introduce to a variety of approaches, from rigorous mathematics to statistics, and from algorithm design and numerical simulations to the theory of phase transitions and to Kosterlitz-Thouless physics and to the interpretation of experiment. It is intended to show how the hard-disk model has continued to shape our view of the physical world, to teach us key aspects, and to prepare for general progress in science.

## Day 1: Hard disks in classical and statistical mechanics (January 24, 2017. First part: Lectures 1-3)

- Introduction

- Overview of this course
- Practical matters
- Models and Theories: A clarification
- Fruit fly "drosophila melanogaster"
- Two-dimensional Ising model
- Hard-disk model

- Molecular dynamics - Newtonian mechanics
- Event-driven Molecular dynamics
- Long-time tails in molecular dynamics
- Algorithms and data structures

- Monte Carlo sampling - Boltzmann mechanics
- Direct sampling
- Markov-chain sampling
- Long-time tails in Monte Carlo dynamics

- Perfect sampling: Coupling from the past, Wilson's algorithm for hard disks

- Statistics of deterministic dynamics
- Determinism and unpredictability

## Day 2: Hard disks and thermodynamical phases (January 26, 2017. First part: Lectures 4-7)

## Day 3: Hard disks and Markov chains (January 31, 2017. Second part: Lectures 1-3)

## Day 4: Hard disks and two-dimensional melting (February 2, 2017. Second part: Lectures 4-7)

## Keywords of the Course

Statistical mechanics, computational physics, phase transitions, colloids, melting transition, Monte Carlo algorithms, Molecular dynamics, Kosterlitz-Thouless transition.

## References

References are accessible via our github server.