In this course you will learn a whole lot of modern physics (classical and quantum) from basic computer programs that you will download, generalize, or write from scratch, discuss, and then hand in. Join in if you are curious (but not necessarily knowledgeable) about algorithms, and about the deep insights into science that you can obtain by the algorithmic approach.

Offered By

## Statistical Mechanics: Algorithms and Computations

École normale supérieure## About this Course

## Offered by

### École normale supérieure

L’École normale supérieure (ENS) est un établissement d'enseignement supérieur pour les études prédoctorales et doctorales (graduate school) et un haut lieu de la recherche française. L'ENS offre à 300 nouveaux étudiants et 200 doctorants chaque année une formation de haut niveau, largement pluridisciplinaire, des humanités et sciences sociales aux sciences dures. Régulièrement distinguée au niveau international, l'ENS a formé 10 médailles Fields et 13 prix Nobel.

## Syllabus - What you will learn from this course

**2 hours to complete**

## Monte Carlo algorithms (Direct sampling, Markov-chain sampling)

Dear students,

**2 hours to complete**

**2 hours to complete**

## Hard disks: From Classical Mechanics to Statistical Mechanics

In Week 2, you will get in touch with the hard-disk model, which was first simulated by Molecular Dynamics in the 1950's. We will describe the difference between direct sampling and Markov-chain sampling, and also study the connection of Monte Carlo and Molecular Dynamics algorithms, that is, the interface between Newtonian mechanics and statistical mechanics. The tutorial includes classical concepts from statistical physics (partition function, virial expansion, ...), and the homework session will show that the equiprobability principle might be more subtle than expected.

**2 hours to complete**

**2 hours to complete**

## Entropic interactions and phase transitions

After the hard disks of Week 2, in Week 3 we switch to clothe-pins aligned on a washing line. This is a great model to learn about the entropic interactions, coming only from statistical-mechanics considerations. In the tutorial you will see an example of a typical situation: Having an exact solution often corresponds to finding a perfect algorithm to sample configurations. Finally, in the homework session we will go back to hard disks, and get a simple evidence of the transition between a liquid and a solid, for a two-dimensional system.

**2 hours to complete**

**2 hours to complete**

## Sampling and integration

In Week 4 we will deepen our understanding of sampling, and its connection with integration, and this will allow us to introduce another pillar of statistical mechanics (after the equiprobability principle): the Maxwell and Boltzmann distributions of velocities and energies. In the homework session, we will push the limits of sampling until we can compute the integral of a sphere... in 200 dimensions!

**2 hours to complete**

## Reviews

### TOP REVIEWS FROM STATISTICAL MECHANICS: ALGORITHMS AND COMPUTATIONS

This is a really good course for the introduction of computational methods in statistical physics. Quite a few topics are covered and very subtle and efficient algorithms are developed and discussed.

Excellent and enthusiastic lectures and tutorials covering a number of topics. Much of the learning took place in the assignments where the concepts were applied and various points were illustrated.

I really enjoyed the course. The only problem was that I was using python 3+ and the programs were written with python 2+. There are some minor differences but I figured the them easily.

Deepen my trust in Monte-Carlo and Markov Chain Monte-Carlo simulation --- exact mimic their analytical counterpart. Also get the chance to touch the spirit of Quantum Mechanics.

## Frequently Asked Questions

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