WASEP
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The document introduces a character named Presley who experiences the dynamics of crowd movement in the London tube, serving as an analogy for the weakly asymmetric simple exclusion process (WASEP). WASEP is described as a stochastic process involving particles that occupy space, with specific emphasis on exclusion principles and transition probabilities that resemble human movement patterns. The document further explores mathematical concepts related to WASEP, including generator systems and hydrodynamic limits, while acknowledging complexities such as large deviations and convergence questions.
WASEP
- 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…