About this Course
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100% online

Start instantly and learn at your own schedule.
Flexible deadlines

Flexible deadlines

Reset deadlines in accordance to your schedule.
Hours to complete

Approx. 54 hours to complete

Suggested: 5 hours/week...
Available languages

English

Subtitles: English...
100% online

100% online

Start instantly and learn at your own schedule.
Flexible deadlines

Flexible deadlines

Reset deadlines in accordance to your schedule.
Hours to complete

Approx. 54 hours to complete

Suggested: 5 hours/week...
Available languages

English

Subtitles: English...

Syllabus - What you will learn from this course

Week
1
Hours to complete
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! <br> 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. <br> 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....
Reading
3 videos (Total 62 min), 2 readings, 2 quizzes
Video3 videos
Tutorial 1: Exponential convergence and the 3x3 pebble game32m
Homework Session 1: From the one-half rule to the bunching method1m
Reading2 readings
Python programs and references10m
Errata (Lecture 1)10m
Quiz1 practice exercise
Practice quiz 1: spotting a correct algorithm4m
Week
2
Hours to complete
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. ...
Reading
3 videos (Total 71 min), 1 reading, 2 quizzes
Video3 videos
Tutorial 2: Equiprobability, partition functions, and virial expansions for hard disks32m
Homework Session 2: Paradoxes of hard-disk simulations in a box2m
Reading1 reading
Python programs and references10m
Quiz1 practice exercise
Practice quiz 2: spotting a correct algorithm (continued)4m
Week
3
Hours to complete
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....
Reading
3 videos (Total 62 min), 2 readings, 2 quizzes
Video3 videos
Tutorial 3: Algorithms, exact solutions, thermodynamic limit31m
Homework Session 3: Two-dimensional liquids and solids2m
Reading2 readings
Python programs and references10m
Errata (Tutorial 3)10m
Quiz1 practice exercise
Practice quiz 3: Spotting a correct algorithm (continued)4m
Week
4
Hours to complete
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! ...
Reading
3 videos (Total 69 min), 1 reading, 2 quizzes
Video3 videos
Tutorial 4: Sampling discrete and one-dimensional distributions34m
Homework Session 4: Sampling and integration in high dimensions2m
Reading1 reading
Python programs and references10m
Quiz1 practice exercise
Practice quiz 4: four disks in a box6m

Instructor

Avatar

Werner Krauth

Directeur de recherches au CNRS
Department of physics

About 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....

Frequently Asked Questions

  • 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.

More questions? Visit the Learner Help Center.