0:28

So with a lattice gas model,

Â the idea is to simulate a gas of particle on a fully discrete system.

Â So you have an example on this picture where you have little arrows which

Â represent molecules, hypothetical molecule with some velocity as shown by this arrow.

Â And at each time step, the particle will move

Â according to the direction of the arrows next to the lattice site.

Â Okay, so he has an example lattice.

Â Of course, I could do that in three dimension,

Â then I would have a cubic lattice.

Â I could also have particle that could move diagonally, that would also be possible.

Â I could also have a lattice which we call

Â hexagonal because it has six possible directions.

Â And the way you build such a lattice is by actually

Â deforming a square lattice with only one diagonal.

Â So what I'm just telling you here, is that you may have

Â different types of spacial structure that you want to use or

Â a different way to decompose your space into cells and

Â basically we'll consider these two example here for historical reason.

Â 1:45

So such a system is called a Lattice gas Automata and

Â the abbreviation is LGA, okay?

Â And as I told you, it's an attempt to have a fully discrete model of fluids.

Â It's a bit similar to what we do in molecular dynamics except that it's really

Â an abstract dynamics and it's discrete in time, in space, and in the state space.

Â 2:14

So it's a representation at some mesoscopic level

Â in which the interaction are very simplified.

Â And I will come back to this in detail,

Â because that's the key of the quality of the model.

Â 2:30

But if you do it right, you can also use some mathematical tools to show that

Â this type of discreet dynamics, they really connect to real systems, okay?

Â We will not do that in detail here, but let me just tell you that

Â you can do some analysis and show whether this type of discrete model,

Â they do or they don't connect to real life.

Â We'll also briefly see that you can describe a gas in movement,

Â but also a diffusion process or chemical reaction or advection processes.

Â So I will spend some time on the description you can do mathematical

Â description, you can do as search system, and

Â of course that directly gives you the way to implement that on a computer.

Â But even though this lattice gas model is of no practical interest,

Â it's really starting point for our next chapter on Lattice Boltzmann's model.

Â So that's why we go in quite a bit of detail, because I think it's intuitively

Â easy to understand how physics work in this created system, and

Â then they will be able to do the step for the Lattice Boltzmann approach.

Â So the idea is that in this finite universe you have a discrete velocity vi,

Â so before we saw it could be left, right, up, and down but

Â according to the lattice you can have a bit more a solution.

Â But the key point is that raising the time,

Â delta t one time step you move always from one lattice point to the next.

Â So there is a clear link between the choice of the velocity that you low in

Â your model and the lattice topology because you need that.

Â Your current position plus your jump is still a lattice point.

Â 4:25

So to describe the system, we introduce what we call occupation number,

Â and we call it n i of r, t can be one or zero.

Â And when it's one it means that you do have a particle entering

Â the lattice site r at time t with velocity i, vi.

Â So this i is the index of the velocity which can be left, up, right, and down,

Â you know simple example.

Â So it mean that you have particle entering when this number's one, and

Â if there's no particle along this channel or direction, then it is zero.

Â So it's really in the learning description.

Â Where you sit on a cell and you look whether something arrives or

Â not in this direction, and at a given time and at a given position.

Â 5:28

sides and direction can be present so

Â really that is occupation number zero or one, it can never be bigger than that.

Â This is formally called exclusion principle but the main benefit for

Â us is that it means that you can always describe your system

Â at any time with a finite number of bits of information.

Â Okay, you will never need to go more than four bits per site a long time.

Â So the first example of Lattice gas is due to Hardy,

Â Pomeau, de Pazzis in 1971, actually it was

Â before the cellular automata was so popular.

Â And the idea was you do exactly what I told you,

Â but you add some collision rules.

Â So it means that particles, they go straight according to the velocity

Â until they meet potentially another particle.

Â So in this first part you see just a free motion particle that goes

Â in a straight line.

Â This situation means two particle with a head on collision.

Â They just meet at the same time, at same position which opposes velocity.

Â And the result of this, is that the particle,

Â the bounds with right angle, okay?

Â That deviated from their horizontal trajectory to a vertical trajectory.

Â And of course, you have the symmetrical situation, where you have a head on

Â collision of the vertical direction and then you go horizontally.

Â And any other configuration you can build out of zero, one, two, three and

Â four particle except these two, they just are left unchanged,

Â meaning that a body can only cross each other without any modification or

Â any actual correlation.

Â 7:23

Why is that so?

Â It means that this rule, they have to implement something which is essential

Â to a gas or fluid which is conservation of mass and momentum.

Â And you can see that here you conserve mass and momentum because you have two

Â particle before collision, you have two particle after collision.

Â You have a zero momentum because they are positive velocity before collision.

Â You have zero momentum after collision because particle has slow

Â positive velocity.

Â 7:50

And so you implement in this interaction, two fundamental rules

Â of a hydrodynamic image which is a mass and momentum conservation.

Â So later on there was another model by Frisch, Hasslacher and

Â Pomeau known as FHP model which correct many of the weakness of this model.

Â I will just illustrate this on this picture.

Â So first, you need a lattice with more symmetry than the square lattice, or

Â we go to the second lattice.

Â And then when you have head-on collision, you have probability to go this way or

Â this way, so you choose that way's probability one-half.

Â In each case, you can also have a three body collision like this which will end

Â up by bouncing the particle this way.

Â And you can also find that this is mass and momentum conserving.

Â [COUGH] For the rest I will focus on

Â the HPP model because it's easier to introduce the math.

Â But first I just would like to show you what happens if you use this FHP model

Â with a lot of particles, and you impose some speed of those particles,

Â let's say from the right boundary, and

Â here you have two obstacle and so this particle we have to move inside.

Â And you see, indeed, some patterns which looks like a fruit.

Â So even though it's discreet particle, very simplified dynamic,

Â if there's enough of those particle.

Â You start seeing something which looks like the emergent of

Â 9:35

Okay, so this is the end of my introduction to lattice gases, and

Â next module we'll detail a little bit the way the interaction

Â between particle can be described mathematically and in the computer.

Â Thank you for your attention.

Â [MUSIC]

Â