Horacio González Duhart23/07/2015

Apparently, this is the last talk before we finish this session…

… and everybody is an astrophysicist or something…

…well, not me. I’m a mathematician…

… that does probability!

But we don’t need to break the momentum… so here’s a planet (or something)

… and there it went.

So let’s go to the other extreme, let’s put some molecules into a container:

What we are going to do is put more particles in a larger container…

And keep going… but maybe we should move a bit further so that we can see the whole picture…

Until a point in which we can no longer distinguish the individual components….

So, the random behaviour of the individual discrete components…

… transforms into a deterministic system of an apparently continuous medium.

Examples of this phenomenon are plenty. Water and other fluid dynamics are just an instance.

Traffic flow

Bacterial growth

Microscopic view Macroscopic view

Or even cats!

Note that a given macroscopic deterministic behaviour may arise from different stochastic microscopic descriptions.

Nothing happening here…

… may be due to some averaging but existing movement.

However…

… it may very well be that there’s nothing at the microscopic level to start with.

How do we find the macroscopic description

from the microscopic stochastic behaviour?

We do maths… we can’t avoid it any longer, we do maths. We call this, the hydrodynamic limit.

The relation between microscopic and macroscopic views

A microscopic behaviour: a stochastic process

A macroscopic description: a differential equation

A Markov process may be completely characterised by 3 objects:

State space: All possible configurations of 0’s and 1’s in a lattice of N sites. (0 means empty site, 1 means occupied site)

Initial configuration: In what configuration of the state space is the process starting?

Infinitesimal generator: This dictates the dynamics of the probabilities of configurations in very small times. We will call this . In this case, particles in the lattice move to the neighbouring sites with equal probability, but there can be at most one particle in each site (think of the cats in the boxes showed previously).

I hate to brag but… we’re talking about some serious maths here and I have not yet written a single equation.Well done, me! Well done!

(Thanks, Mr. Putin)

A deterministic dynamical system may be completely characterised by a partial differential equation…

This is the PDE

… and initial and boundary conditions.

Space

Densityr

At time t=0:

u

So when we say that…

We actually mean that

Average of the microscopic configurations

Average of the density in macroscopic space

Probability of Very small 1

If there are a lot of particles

Graphically…

Space

Densityr

u

Low density where there are few particles

High density where there are many particles

Told you, maths wasn’t that scary after all…