This is a talk given at UCL for the XIII Symposium of Mexican Students and Studies.

1. Horacio González Duhart
23/07/2015

2. 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.

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

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

5. 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….

6. So, the random behaviour of the individual discrete components…
… transforms into a deterministic system of an apparently continuous medium.

7. Examples of this phenomenon are plenty. Water and other fluid dynamics are just an instance.
Traffic flow
Bacterial
growth
Microscopic view Macroscopic view

8. Or even cats!

9. 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.

10. 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.

13. The relation between
microscopic and
macroscopic views
A microscopic
behaviour: a stochastic
process
A macroscopic
description: a
differential equation

14. 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 𝜇 𝑁(𝑡). 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).

15. 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)

17. A deterministic dynamical system may be completely characterised by a partial differential equation…
This is the PDE
… and initial and boundary conditions.
Space
Density
r
At time t=0:
u

18. 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

19. Graphically…
Space
Density
r
u
Low density where there
are few particles
High density where there
are many particles

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