I gave this talk for the PSS at the University of Bath on the 9th of October. As usual, all the animations can't be seen on this file.

1. Horacio González Duhart
hgd20@bath.ac.uk

2. Meet Presley
He comes from a small ex Soviet country (there are loads of them, but this one is really
small)
So one day, Presley takes the tube in London!

3. So when he’s inside… he realises he’s not alone...
(imagine this is a closed door)
… and waits for the next stop to get down.

4. So, the door opens and he tries to get out…
Following the dynamics of
Presley or any of the respectable
law abiding London citizens is a
NOOB! nearly impossible task due to all
the pushing around.

5. However, what the security guard sees on the CCTV screen is…
He doesn’t see the
pushing, he can’t even
recognise Presley from
the rest and it seems like
everybody is following a
clear pattern!

6. A word on Statistical mechanics
Stochastic processes
Stochastic model
Microscopic scale
Macroscopic scale
Deterministic model
Partial differential equations

7. So, we’re supposed to talk about the WASEP? What is the WASEP? Why is the previous
story relevant? In short… WTF?
WASEP stands for
Weakly Asymmetric Simple Exclusion Process
Maybe just the words do not help comprehending the concept, so we’ll explain word by
word… in reversed order.

8. Weakly Asymmetric Simple Exclusion Process
The word process, refers to a stochastic process.
In fact, our process is actually a Markov process in continuous time.
Talking about Stochastic processes in general is just saying too much, so we’ll see a more
detailed definition of our process further in the presentation. Meanwhile we just need to
know that it is a system of particles in a certain space that changes in time in a random
way, pretty much like individuals moving around in the tube.

9. Weakly Asymmetric Simple Exclusion Process
The word exclusion is the key concept of this whole show. Exclusion means, that no more
than one particle can occupy a spot on the space… still just as people in the tube.

10. Weakly Asymmetric Simple Exclusion Process
Simple is the key word that allows to work on an “easy” setting rather than on a “hard”
one. The time for writing some definitions has arrived.
The state space will be , so for example an element from this space
may look like
We’ll call “configurations” to these elements.
We’ll denote our process , so at every time we have a
random configuration

11. Now, how do we change from one configuration to another in a random way?
For that we need some exponential clocks…
What is an “exponential clock”?
A funny way of thinking of an exponential clock with rate l, would be of a wire walker
on a rope at height l, and he suffers from acrophobia… and we time how long he takes
to fall down, when this happens the clock rings.
So if the rate is high we expect the
clock to ring fast! When the rate is
slow, the wire walker won’t be afraid
and stay on the wire for longer.

12. So, it will work like this. Below each site we put an exponential clock (all with the same
rate and independent from each other). When a clock rings, if there is a particle at that
site then it moves with probability p to the right or with probability q=1-p to the left.
However, recall the exclusion property, so if there is already a particle where this one
wants to move, then the move is cancelled, and we restart the clock.
Nothing happens because there’s already a particle in the right side.
Assume p=1, so the particle moves to the right.
Later…

13. The process is called simple because particles can only jump to either the left or right
immediate neighbour.
However one might define probabilities of particles moving from one site to any
other.
One can also define these kind of processes in a multidimensional lattice.

14. Weakly Asymmetric Simple Exclusion Process
In literature, one can find 4 kinds of simple exclusion processes:
Symmetric, or SSEP, when the probability of a particle jumping to the right is the same
as jumping to the left (and hence equal to ½).
Asymmetric, or ASEP… yes! You guessed! When the probabilities are not the same.
However, we call totally asymmetric, or TASEP, in which one of the probabilities is
equal to 1.

15. Weakly Asymmetric Simple Exclusion Process
The fourth one is the weakly asymmetric, or WASEP. To explain this process we
need to change the process a bit.
First take a natural number , and now our state space will be
where
Now, fix a number , and let the probability of moving to the right depend
on this number and the number of sites like this:

16. Weakly Asymmetric Simple Exclusion Process
Formally speaking, the infinitesimal generator for the WASEP:
where
Diffusive
scaling
Boundary conditions at sites –N and N

17. Weakly Asymmetric Simple Exclusion Process
For a fixed N, this is an ASEP actually on a finite state space, hence the question still
stand, why weakly?
We will let N go to infinity and so…
On this side of the limit , the probability of jumping to the right looks pretty much
symmetric... Yet the process never was, and hence weakly asymmetric.

18. Now what?
Well, as any other Markov process we would be interested if it has an invariant measure…
is it unique? How fast can we reach these invariant measures? Is it a reversible process?
All these interesting questions have an answer… yet, we’ll focus in the hydrodynamic
limit.

19. I believe the easiest case for finding the hydrodynamic limit is the TASEP (with p=1)
It is so easy, that here I present you the…
First Power Point proof of the TASEP’s
hydrodynamic limit
Microscopic scale Macroscopic scale
0
0
0
0
0
0

20. Fair enough, it’s not an actual proof… but hopefully the idea is convincing enough.
Formally,
The empirical density measure:
Converges weakly to a measure
given by a function r
Beware of the cases! (the time scaling for the ASEP is
hyperbolic rather than diffusive)
That satisfies the PDE:
ASEP (Burger’s equation)
WASEP (Viscous Burger’s equation)
SSEP (Diffusion equation)
There’s a half missing here, I just noticed this morning…

21. To summarise…
Hydrodynamic limit
WASEP
But there’s more when incorporating
large deviations!
Rate functional
And even more if we try finding a limit
in the gamma sense!
G-convergence
Questions such as “what are large deviations?” or “what is gamma convergence?” are
very legitimate ones… however they deserve a presentation on their own…

22. No ex soviet national was hurt during the preparation of these slides…
Thanks for your time…