list 10 sequences
assignment Level : Introductive
label Physics
chat_bubble_outline Language : English
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credit_card Free access
verified_user Fee-based Certificate
timer 40 hours in total

About the content

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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  • Week 1 - Monte Carlo algorithms (Direct sampling, Markov-chain sampling)
    Dear students, welcome to the first week of Statistical Mechanics: Algorithms and Computations!
    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 o...
  • Week 2 - 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 Molecu...
  • Week 3 - 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 ty...
  • Week 4 - 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 velocitie...
  • Week 5 - Density matrices and Path integrals (Quantum Statistical mechanics 1/3)
    Week 5 is the first episode of a three-weeks journey through quantum statistical mechanics. We will start by learning about density matrices and path integrals, fascinating tools to study quantum systems. In many cases, the Trotter approximation will be useful...
  • Week 6 - Lévy Quantum Paths (Quantum Statistical mechanics 2/3)
    In Week 6, the second quantum week, we will introduce the properties of bosons, indistinguishable particles with peculiar statistics. At the same time, we will also go further by learning a powerful sampling algorithm, the Lévy construction, and in the homewor...
  • Week 7 - Bose-Einstein condensation (Quantum Statistical mechanics 3/3)
    At the end of our quantum journey, in Week 7, we discuss the Bose-Einstein condensation phenomenon, theoretically predicted in the 1920's and observed in the 1990's in experiments with ultracold atoms. In the path-integral framework, an elegant description of ...
  • Week 8 - Ising model - Enumerations and Monte Carlo algorithms
    In Week 8 we come back to classical physics, and in particular to the Ising model, which captures the essential physics of a set of magnetic spins. This is also a fundamental model for the development of sampling algorithms, and we will see different approache...
  • Week 9 - Dynamic Monte Carlo, simulated annealing
    Continuing with simple models for spins, in Week 9 we start by learning about a dynamic Monte Carlo algorithm which runs faster than the clock. This is easily devised for a single-spin system, and can also be generalized to the full Ising model from Week 8. In...
  • Week 10 - The Alpha and the Omega of Monte Carlo, Review, Party
    The lecture of Week 10 includes the alpha and the omega of our course. First we repeat the experiment of Buffon's needle, already performed in the 18th century, and then we touch the sophisticated theory of Lévy stable distributions, and their connection with ...


Werner Krauth
Directeur de recherches au CNRS
Department of physics


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École normale supérieure

The École normale supérieure (ENS) is an elite higher education institution (graduate school) for advanced undergraduate and graduate studies, and a prestigious French research center. It encompasses fifteen teaching and research departments, spanning the main humanities, sciences and disciplines. Unique among France’s grandes écoles for its training in the humanities and sciences, the ENS prepares its students for their role as future leaders in every imaginable professional field: in research, media, public service and private industry.




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