# MOOC SMAC

### From Werner KRAUTH

## Contents |

# 2014 Edition

The Massive open online course (MOOC) on Coursera: Statistical Mechanics: Algorithms and Computations, that started on February 3rd, 2014, and drew 30,000 registered students, is now finished. The last of the ten courses was published on April 13, 2014, and the final exam ran until 14 May, 2014. The course videos were viewed 250,000 times, thousands of homeworks were handed in, and there were about 5200 (!) entries on the forum that were viewed far more than 120,000 times, all of this of course for free.

The story of our MOOC was written up in an editorial "Coming home from a MOOC" for "Computing in Science & Engineering" (2015).

# 2015 Edition

The 2nd edition of the Massive open online course (MOOC) on Coursera: Statistical Mechanics: Algorithms and Computations ran from February 2nd, 2015 through May 26, 2015. Again a great experience, with a new enthousiastic crowd of students. Leave feedback on Twitter under the hashtag #SmacMOOC2015.

# 2016 Edition - self-paced

The 3rd edition of SMAC comes with two major change:

- Statistical Mechanics: Algorithms and Computations is now a self-paced course, just like all other courses on Coursera. This means that after the course that started on 29 February, 2016, another cohort will start in March 2016, then in April 2016. If students lose track at some moment, they will be able to continue later. I will be curious to see how it will turn out, especially whether the individual pace still allows some kind of group experience. In any case, we put in a lot of effort to make our popular course accessible to an even larger community of students. We will continue to be very present on the forum! So let's all have fun with the third edition of SMAC.
- There will be no more certificate, as ENS was unable to keep it free of charge. However, students at some Universities have taken this course for credit (at their Universities), following special arrangements.

# Syllabus

Here is a list of the topics of the ten weeks. For each week, you can also find the sections of the SMAC text book where useful material can be found (NB: SMAC = W. Krauth - Statistical Mechanics: Algorithms and Computations, Oxford 2006)

## WEEK 1

Lecture: Introduction to Monte Carlo algorithms

Tutorial: Exponential convergence and the 3x3 pebble game

Homework: From the one-half rule to the bunching method (peer-graded)

Relevant SMAC sections: 1.1.1 (children game - direct sampling), 1.1.2 (adults game - Markov-chain sampling), 1.1.4 (pebble game, detailed balance and transfer matrix), 1.3.5 (error estimates and bunching), 1.4.1 (ergodicity)

## WEEK 2

Lecture: Hard disks: from Classical Mechanics to Statistical Mechanics

Tutorial: Equiprobability, partition functions, and virial expansions for hard disks

Homework: Paradoxes of hard-disk simulations in a box (peer-graded)

Relevant SMAC sections: 2.1.1 (molecular dynamics), 2.2 (equiprobability), 2.2.1 (direct sampling for hard disks), 2.2.2 (partition function, acceptance rate, virial expansion), 2.2.3 (Markov-chain sampling for hard disks).

## WEEK 3

Lecture: Energy, free energy, and phase transitions

Tutorial: Entropic interactions and the Random Clothes-pins model

Homework: Two-dimensional phase transitions (peer-graded)

Relevant SMAC sections: 6.1.1 (random clothes-pin), 6.1.2 (depletion interaction) Recreational multiple choice: Spotting a correct algorithm

## WEEK 4

Lecture: Sampling and Integration - From Gaussians to the Maxwell and Boltzmann distributions

Tutorial: Sampling discrete and one-dimensional distributions

Homework: Sampling and integration in high dimensions (peer-graded)

Relevant SMAC sections: 1.2.1 (random number generator), 1.2.3 (discrete distributions, tower sampling), 1.2.4 (connection between sampling and integration), 1.2.5 (sampling a Gaussian), 1.2.6 (sampling a sphere or its surface), 2.2.4 (Maxwell distribution of velocities).

## WEEK 5

Lecture: Density matrices and path integrals

Tutorial: Trotter decomposition and quantum time-evolution

Homework: Quantum statistical mechanics and Quantum Monte Carlo (peer-graded)

Relevant SMAC sections: 3.1.1 (quantum harmonic oscillator, wave functions, energy levels, density matrix), 3.1.2 (density matrix in free space and in a periodic box), 3.1.3 (density matrix in a box with periodic/open boundary conditions), 3.2.1 (Trotter formula, convolution of density matrices), 3.3 (path-integral formulation), 3.3.1 (path integral Monte Carlo)

## WEEK 6

Lecture: Lévy sampling of quantum paths

Tutorial: Bosonic statistics (with wave functions)

Homework session 6: Path sampling: A firework of algorithms (peer-graded)

Relevant SMAC sections: 3.3.2 (Levy construction in free space and in the harmonic trap), 4.1.1 (density of states), 4.1.2 (bosonic statistics, energy of N trapped bosons, condensed fraction)

## WEEK 7

Lecture: Quantum indiscernability and Bose-Einstein condensation

Tutorial: Permutation cycles and ideal Bosons

Homework: Bosons in a trap - Bose-Einstein condensation (peer-graded)

Relevant SMAC sections: 4.2.1 (density matrix for bosons), 4.2.2 (permutation cycles counting, recursion formula for the partition function), 4.2.4 (condensate fraction, probability of cycle lengths), 4.2.5 (direct sampling algorithm for ideal bosons in a trap)

## WEEK 8

Lecture: Ising model - From enumeration to Cluster Monte Carlo Simulations

Tutorial: Ising model - Heat bath algorithm, coupling of Markov chains

Homework: Cluster sampling, perfect sampling in the Ising model (peer-graded)

Relevant SMAC sections: 5.1 (Ising model), 5.1.1 (enumeration of states), 5.1.2 (thermodynamics of small systems), 5.2 (Monte Carlo sampling), 5.2.1 (local algorithm), 5.2.2 (heat-bath algorithm and perfect sampling), 5.2.3 (cluster algorithm)

## WEEK 9

Lecture: Dynamic Monte Carlo methods and the faster-than-the-clock approach

Tutorial: Simulated annealing and the 13-sphere problem

Homework: Simulating annealing for sphere packings and the Traveling salesman (peer-graded)

Relevant SMAC sections: 7.2.1 (faster-than-the-clock sampling for one spin), 7.3 (disks on a sphere), 7.3.1 (simulated annealing)

## WEEK 10

Lecture: The alpha and omega of Monte Carlo algorithms (Buffon's needles and Lévy's stable distributions)

Tutorial: Review, Summary, Outlook

Relevant SMAC sections: 1.1.3 (Buffon's needle), 1.4.2 (gamma-integral), 1.4.4 (stable distributions)