Hello, bonjour,
welcome to the tenth tutorial of Statistical Mechanics: Algorithms and Computations
from the Physics Department of Ecole Normale Superieure.
We have now come to the end of a long journey
and after almost three months of travels
(and of struggles)
time has come to revisit the landmark sites
that we have seen during the last two months and a half.
In week 1, our mascot was this pebble.
We used it on the Monte Carlo beach
and on the heliport of Monaco,
as well as in the homogeneous and inhomogeneous
3x3 pebble game.
It illustrated such essential concepts as detailed balance,
aperiodicity,
and irreducibility, as well as the Metropolis algorithm.
In the homework session, it came up in the 1/2 rule, that we used many times since.
So, please remember the pebble from week 1.
In week 2 we moved from pebbles to disks.
This gave us the opportunity to discuss about molecular dynamics
and Monte Carlo methods.
We were plunged in deep discussions and questions
in physics: the equivalence between Newton dynamics
and the Boltzmann statistical approach.
This is one of the big mysteries in physics, the so-called ergodic hypothesis,
that you actually proved to be valid for hard disks in homework 2
using Monte Carlo methods (direct sampling and Markov chain) and molecular dynamics.
During the third week, we went one step further and turned our attention
to phase transitions, which are one of the main subjects of statistical mechanics.
With the help of this object, the clothes-pin, you have studied the case of one-dimensional hard spheres,
and you realized that by conceiving a direct rejection-free algorithm
it was also possible to compute the partition function and to show that the system presents no phase transition.
Following the same way, you have also understood the origin of the famous Asakura-Oosawa depletion interaction.
In the homework, you turned your attention to the case of two-dimensional hard disks,
and you understood how this very same interaction
was responsible for the existence of phase transition in this system,
as famously shown by Alder and Wainwright in 1964.
Week 4 was dedicated to the relation between integration and sampling.
We moved from the sampling of points on the surface of a sphere,
like this basket-ball here, to the Maxwell distribution of velocities in a gas
and the Boltzmann distribution of energies,
and also to essential concepts like discrete sampling and sampling transformations.
In the homework we performed integration and sampling in no less than 200 dimensions.
During week 5, you turned your attention to the case of quantum particles,
you learned to deal with their statistics using the tool of the density matrix.
Do you remember this slinky? It embodies the path integral
that you could construct using three simple properties of the density matrix:
the convolution property, the Trotter decomposition and the solution for the free particle.
With these tools in your hands, you learned how to construct
- using a sampling approach -
quantum Monte Carlo algorithms, to evaluate numerically the density matrix.
During the tutorial, you also learned how to perform the imaginary time rotation
and to study the dynamics of one particle in one dimension.
Week 6 was also dedicated to quantum mechanics:
we introduced the Levy quantum path,
a direct sampling Monte Carlo method for path integral.
You used this method in the homework, both in its free and harmonic versions.
In the tutorial we started to discuss about bosons
and Bose-Einstein condensation, which takes place in
experimental cells like this one.
Week 7 was all about Bose-Einstein condensation, as it takes place in Alberto's experimental cell.
It can be looked at and manipulated by lasers, so I hope you all wore your glasses.
But seriously, it was about a true quantum phenomenon: the indistinguishability of bosons
and about our glorious 40-lines program: harmonic_bosons.py.
In the homework, we dealt with bosons in traps shaped like cigars, and even pancakes.
This sounds like fun, and it was a lot of fun!
But it was also a serious non-trivial quantum mechanics problem.
In week 8, we returned to classical physics
and to the Ising model of spins on a lattice
with a rudimentary nearest neighbors interaction.
From exact enumeration of plus and minus spins,
using the Gray code, we moved on to a discussion of local Monte Carlo methods
using the Metropolis and the heat-bath algorithm
and to the fabulous cluster methods.
And do you remember the story of coupling?
During week 9 we went on discussing our beloved model of statistical mechanics
In many of these models, events occur at random times
and during the lecture we learned how to sample the times between these events
without having to wait for them to occur.
And in this way we constructed faster-than-the-clock algorithms.
During the tutorial, you were introduced to a basic but very interesting
strategy, the simulated annealing, that allows to deal with complex optimization problems.
For instance we have studied the case of the 13-spheres problem
and we were able to devise an algorithm which finds the best solution to this problem in a few seconds.
So, finally, week 10
was the alpha and the omega of Monte Carlo.
The alpha was where all started,
in 1777 with Buffon's needles
thrown randomly onto a parquet,
ant the omega was where all ended
with the famous Levy stable distributions
for observables with infinite variance.
So, this was it, Statistical Mechanics: Algorithms and Computations.
There were many subjects that we left out,
but you will agree that there was a lot of stuff inside.
So now it is time to present the members of the team who where not in front of the camera during all this time
So let's start with Maxim Berman..
Hi everybody, I had a lot of fun producing the best animations possible for the course
including the histograms, the cascades of configurations and the pebbles thrown on the Monte Carlo beach.
Then there is Tommaso Comparin..
Ciao, it's me who looked after many of the programs
and also typed the subtitles for the course, including the ones you are reading now.
I hope this turned out useful for you!
Then there is Emilie Noblet..
So hello everybody, I was the camera woman on this project,
entirely shot in the green-screen technology.
In addition there was more camera by Frederic Borjat,
and our editor Baptiste Ribrault,
who put all these pieces so nicely together.
So thanks to all of you, all over the world
for your interest and your active participation:
*you* made this course happen.
And now, let's have a party!