Hello, bonjour, gamardjoba, jo napo,
welcome to the fourth homework session of Statistical Mechanics: Algorithms and Computations
from the Physics Department of Ecole Normale supérieure.
In this week's lecture and tutorial, we really understood
the connection between sampling and integration.
Let us now use what we learned to calculate difficult,
high-dimensional integrals,
a field where Monte Carlo methods just cannot be beaten.
The central object this week was the unit hypersphere,
let's compute its volume using a Markov-chain Monte Carlo random walk in high dimensions.
These algorithms are extremely efficient,
as you will find out for yourselves
and they allow for beautiful calculations.
On the Monte Carlo beach, it is impossible to compute the area of the circle,
that is to say of a two-dimensional sphere, in a direct way.
Instead, what we can determine is the ratio between the area of the circle and the area of the square.
In this week's homework, you will face a similar problem:
it is impossible to determine directly the volume of sphere in d dimensions.
Instead, what you will write is a program that allows you to determine the ratio
between the volumes of spheres in successive dimensions: d, d+1.
All in all, by accumulating results, you will be able to determine the volume of a sphere
not in 3 dimensions, but in 200 dimensions
with a tight control on the error.
You will get a beautiful result, which illustrates the power of statistical mechanics in the design of algorithms.