Statistical Mechanics: Algorithms and Computations

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Statistical Mechanics: Algorithms and Computations

École normale supérieure

About this Course

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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.

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Start instantly and learn at your own schedule.

Flexible deadlines

Reset deadlines in accordance to your schedule.

Approx. 54 hours to complete

Suggested: 5 hours/week...

English

Subtitles: English...

100% online

Start instantly and learn at your own schedule.

Flexible deadlines

Reset deadlines in accordance to your schedule.

Approx. 54 hours to complete

Suggested: 5 hours/week...

English

Subtitles: English...

Syllabus - What you will learn from this course

Week

1

1 hour to complete

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

Dear students,
welcome to the first week of Statistical Mechanics: Algorithms and Computations!
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Here are a few details about the structure of the course: For each week, a lecture and a tutorial videos will be presented, together with a downloadable copy of all the relevant python programs mentioned in the videos. Some in-video questions and practice quizzes will help you to review the material, with no effect on the final grade. A mandatory peer-graded assignment is also present, for weeks from 1 to 9, and it will expand on the lectures' topics, letting you reach a deeper understanding. The nine peer-graded assignments will make up for 50% of the grade, while the other half will come from a final exam, after the last lecture.
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In this first week, we will learn about algorithms by playing with a pebble on the Monte Carlo beach and at the Monaco heliport. In the tutorial we will use the 3x3 pebble game to understand the essential concepts of Monte Carlo techniques (detailed balance, irreducibility, and a-periodicity), and meet the celebrated Metropolis algorithm. Finally, the homework session will let you understand some useful aspects of Markov-chain Monte Carlo, related to convergence and error estimations....

Tutorial 1: Exponential convergence and the 3x3 pebble game32m

Homework Session 1: From the one-half rule to the bunching method1m

2 readings

Python programs and references10m

Errata (Lecture 1)10m

1 practice exercise

Practice quiz 1: spotting a correct algorithm4m

Week

2

1 hour 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.
...

Tutorial 2: Equiprobability, partition functions, and virial expansions for hard disks32m

Homework Session 2: Paradoxes of hard-disk simulations in a box2m

1 reading

Python programs and references10m

1 practice exercise

Practice quiz 2: spotting a correct algorithm (continued)4m

Week

3

1 hour 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....

Homework Session 3: Two-dimensional liquids and solids2m

2 readings

Python programs and references10m

Errata (Tutorial 3)10m

1 practice exercise

Practice quiz 3: Spotting a correct algorithm (continued)4m

Week

4

1 hour 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!
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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....

Frequently Asked Questions

When will I have access to the lectures and assignments?

Once you enroll for a Certificate, you’ll have access to all videos, quizzes, and programming assignments (if applicable). Peer review assignments can only be submitted and reviewed once your session has begun. If you choose to explore the course without purchasing, you may not be able to access certain assignments.