[MUSIC] So welcome back to our course on simulation and modeling of natural processes. So we continue our chapter on cellular automata modeling. We start with a module on lattice-gas models. So with a lattice gas model, the idea is to simulate a gas of particle on a fully discrete system. So you have an example on this picture where you have little arrows which represent molecules, hypothetical molecule with some velocity as shown by this arrow. And at each time step, the particle will move according to the direction of the arrows next to the lattice site. Okay, so he has an example lattice. Of course, I could do that in three dimension, then I would have a cubic lattice. I could also have particle that could move diagonally, that would also be possible. I could also have a lattice which we call hexagonal because it has six possible directions. And the way you build such a lattice is by actually deforming a square lattice with only one diagonal. So what I'm just telling you here, is that you may have different types of spacial structure that you want to use or a different way to decompose your space into cells and basically we'll consider these two example here for historical reason. So such a system is called a Lattice gas Automata and the abbreviation is LGA, okay? And as I told you, it's an attempt to have a fully discrete model of fluids. It's a bit similar to what we do in molecular dynamics except that it's really an abstract dynamics and it's discrete in time, in space, and in the state space. So it's a representation at some mesoscopic level in which the interaction are very simplified. And I will come back to this in detail, because that's the key of the quality of the model. But if you do it right, you can also use some mathematical tools to show that this type of discreet dynamics, they really connect to real systems, okay? We will not do that in detail here, but let me just tell you that you can do some analysis and show whether this type of discrete model, they do or they don't connect to real life. We'll also briefly see that you can describe a gas in movement, but also a diffusion process or chemical reaction or advection processes. So I will spend some time on the description you can do mathematical description, you can do as search system, and of course that directly gives you the way to implement that on a computer. But even though this lattice gas model is of no practical interest, it's really starting point for our next chapter on Lattice Boltzmann's model. So that's why we go in quite a bit of detail, because I think it's intuitively easy to understand how physics work in this created system, and then they will be able to do the step for the Lattice Boltzmann approach. So the idea is that in this finite universe you have a discrete velocity vi, so before we saw it could be left, right, up, and down but according to the lattice you can have a bit more a solution. But the key point is that raising the time, delta t one time step you move always from one lattice point to the next. So there is a clear link between the choice of the velocity that you low in your model and the lattice topology because you need that. Your current position plus your jump is still a lattice point. So to describe the system, we introduce what we call occupation number, and we call it n i of r, t can be one or zero. And when it's one it means that you do have a particle entering the lattice site r at time t with velocity i, vi. So this i is the index of the velocity which can be left, up, right, and down, you know simple example. So it mean that you have particle entering when this number's one, and if there's no particle along this channel or direction, then it is zero. So it's really in the learning description. Where you sit on a cell and you look whether something arrives or not in this direction, and at a given time and at a given position. So to make this easy to implement in a computer, we ask that no more than one particle per sides and direction can be present so really that is occupation number zero or one, it can never be bigger than that. This is formally called exclusion principle but the main benefit for us is that it means that you can always describe your system at any time with a finite number of bits of information. Okay, you will never need to go more than four bits per site a long time. So the first example of Lattice gas is due to Hardy, Pomeau, de Pazzis in 1971, actually it was before the cellular automata was so popular. And the idea was you do exactly what I told you, but you add some collision rules. So it means that particles, they go straight according to the velocity until they meet potentially another particle. So in this first part you see just a free motion particle that goes in a straight line. This situation means two particle with a head on collision. They just meet at the same time, at same position which opposes velocity. And the result of this, is that the particle, the bounds with right angle, okay? That deviated from their horizontal trajectory to a vertical trajectory. And of course, you have the symmetrical situation, where you have a head on collision of the vertical direction and then you go horizontally. And any other configuration you can build out of zero, one, two, three and four particle except these two, they just are left unchanged, meaning that a body can only cross each other without any modification or any actual correlation. Why is that so? It means that this rule, they have to implement something which is essential to a gas or fluid which is conservation of mass and momentum. And you can see that here you conserve mass and momentum because you have two particle before collision, you have two particle after collision. You have a zero momentum because they are positive velocity before collision. You have zero momentum after collision because particle has slow positive velocity. And so you implement in this interaction, two fundamental rules of a hydrodynamic image which is a mass and momentum conservation. So later on there was another model by Frisch, Hasslacher and Pomeau known as FHP model which correct many of the weakness of this model. I will just illustrate this on this picture. So first, you need a lattice with more symmetry than the square lattice, or we go to the second lattice. And then when you have head-on collision, you have probability to go this way or this way, so you choose that way's probability one-half. In each case, you can also have a three body collision like this which will end up by bouncing the particle this way. And you can also find that this is mass and momentum conserving. [COUGH] For the rest I will focus on the HPP model because it's easier to introduce the math. But first I just would like to show you what happens if you use this FHP model with a lot of particles, and you impose some speed of those particles, let's say from the right boundary, and here you have two obstacle and so this particle we have to move inside. And you see, indeed, some patterns which looks like a fruit. So even though it's discreet particle, very simplified dynamic, if there's enough of those particle. You start seeing something which looks like the emergent of the structure at the large scale like eddies or turbulence and things like that. So it shows that really there's something to do with fluid dynamics. Okay, so this is the end of my introduction to lattice gases, and next module we'll detail a little bit the way the interaction between particle can be described mathematically and in the computer. Thank you for your attention. [MUSIC]