0:24

start this module on microdynamics of lattice gasses.

Â So I would like to know how we can describe the detailed

Â movement of our particle in a gas.

Â So you remember that these particle they are moving in a regular lattice with

Â some discrete possible velocity.

Â 0:43

And when they hit each other there's a collision, and so

Â they take another direction.

Â So this is illustrated in this slide.

Â Again, where you can see here, several particles meeting at the same place,

Â colliding, then going out with a new direction and then moving to the nearest

Â neighbor where more particles will be met and new collisions will happen.

Â So, this picture tells you that you can actually divide the process two sub steps,

Â one is the collision, and one is the propagation or streaming step.

Â So [COUGH] last time we introduced this occupation number and

Â I telling us where there is or not a particle with velocity,

Â v i entering the side at the given time.

Â So now we just put another subscript which is in or

Â out to distinguish between the particle entering from the particle going out.

Â And then after you go out you are moving into the nearest neighbor side, okay?

Â So with this, you can formulate

Â the microdynamics of this particle with these two equations.

Â So first you get a new distribution of out going particle

Â from the in going particle plus a collision term which are actually

Â interaction of all the particle at this position.

Â And then this propagation telling you that If you are out

Â with a velocity in direction i at position r and t,

Â you will be in with still a velocity in the same direction i.

Â But on the nearest neighbor along that direction v and i,

Â at the next time slot, okay?

Â So, basically that's the two equations that describe the system, and

Â all the collision process is hidden in this function that we will

Â express in a few moments, okay?

Â So of course you can combine these two equation in only one and

Â you get what is the very well known equation of evolution of the population

Â at [COUGH] every lattice point.

Â Where we just used this notation that when you don't put any subscript,

Â it mean that it's actually an incoming particle.

Â 3:17

And if the collision term would be zero,

Â it would mean that just we have a free streaming of the particle.

Â It'd just keep moving straight.

Â But that wouldn't be a very interesting problem.

Â So, of course, we want this collision term to do something interesting.

Â 3:31

So I remind you that for this HPP gas, which is probably the simplest

Â we can consider, there are two important collision, is these two here which is

Â two hadron collision, which mean that if a particle in the direction, let's say i.

Â 3:50

Meets a particle in the direction i plus two, meaning just the opposite,

Â we have four directions.

Â So I plus two is the opposite direction of I.

Â Okay, it will create new particles in the direction i plus 1 and i plus 3,

Â okay, so that's basically what we want to formulate in our equation.

Â So for this HPP model I have only four possible

Â velocity which in a Cartesian coordinate are those, and

Â then this is the expression of what I was just telling you in the previous slides.

Â Saying that the outgoing particle in direction i is one or zero

Â and if it were freeze framing it would be the same as the incoming particle.

Â So if this one is 1, this one is 1 too, this one is 0, this one is 0 too.

Â But now suppose it's 1.

Â So this particle can be deviated to another channel, another direction

Â provided you have exactly a particle, i + 2 in the opposite direction.

Â And no other particle in the vertical direction.

Â So in that case, you will destroy this particle in the direction i and you will

Â on the other hand create a particle in the other direction perpendicular to it.

Â So you can that you do it yourself,

Â quietly show that this expression is a Boolean expression,

Â so the quantity can be 0 or 1.

Â And exactly reproduce the collision scheme that I show you in the previous

Â slide, okay.

Â 5:31

So, now we would like to make the connection with physics, and

Â in physics we like to talk about density and velocity.

Â So, what is the density?

Â Well, it's the sum of all the particles that arrive at the same position.

Â So if you sum up for the four direction, the particle is enters

Â the site on the side t, you get the density of the number of particles,

Â and that even site, okay?

Â And actually you can show that the outgoing mass,

Â which is the sum of all the particles that leave that site,

Â is exactly conserved by the rule.

Â And this is a property of collision terms, so

Â you can explicitly compute that from this expression by

Â summing this over i, and you will see that most of the terms, they cancel out.

Â 6:31

You can also compute the momentum of a site, which we call j.

Â Which is by definition the product of the density

Â by the velocity field of that site.

Â And it's just the sum of the momentum of all the particles.

Â So each particle has a momentum in the direction of its velocity.

Â So in case of mass equal 1,

Â this is just the sum of the momentum of all the particle entering the side, okay?

Â And that can give me quantity j, but

Â I also got the quantity rho from my previous sum, so

Â I can extract the velocity associated to that specific cell by dividing j by rho.

Â 7:34

So, I'd like to show some demos and

Â maybe give you some information about this demo.

Â And the first one

Â I'd like to show you is how does it look like this HPP gas.

Â When you simulate that, and

Â the my accumulation will be to have a box with a lot of particles.

Â And in the middle of this box I will put a higher density of particle.

Â So will see that mostly the particle are uniformly distributed over space.

Â Except in the middle, would mean that I put a high density in my gas,

Â and this high density will propagate as a sound wave if you want a pressure wave.

Â So we'll run that, and we'll see that of course you see this wave going on.

Â 8:28

If I run it a second time you see that the high density I had

Â in the middle propagates away as a wave.

Â But it has a very weird shape.

Â Okay, you will expect a round shape for

Â a wave and here you see more like a square or a diamond shape.

Â And that's of course a sign that perhaps something is not optimal in this model.

Â So if you do the same now with this FHP model which has more symmetry,

Â you see that things are going much better.

Â So you still see a slight hexagonal structure.

Â But if you're far enough from the center, you are also very close to a circle.

Â And you can show that to second order of accuracy, this is a perfect circle.

Â So you can see it slightly as an ellipse because when you do

Â hexagonal lattice on a computer, you have to do some deformation, and

Â then the horizontal and vertical axes are not the same scale.

Â That's why you have a slight deformation which is truly artificial in the picture.

Â But still you see that by adding more symmetry to your space

Â you get also a much better symmetry of your macroscopic imaging behavior.

Â 9:51

Now what's interesting also with these lattice gases is that they

Â can compute the movement of a lot of particle in a totally exact way,

Â there is no rounding error, no loss of information.

Â So this is illustrated with this experiment.

Â So you see, this is my gas HPP, so

Â is this a square symmetry and you could see a very artificial way

Â during how the particle which were continuous the left part of the system,

Â they start to invade the empty right compartment.

Â They can only move horizontally until they start hitting the wall and then mixing.

Â But still, what I'm claiming here is that the dynamic I've obtained here

Â is totally deterministic without any errors, meaning that

Â actually I could use this property of physics, which is time reversibility.

Â Know that the equation of Newton, we are invariant by reversing time,

Â if you reverse the velocity the particles.

Â Here it means that if I take all the particles, and

Â those going left, I told them to go right.

Â Those going up, I told to go down, and so on.

Â And I just run the exact same rule while I should actually retrace my own past.

Â Okay, that's the time we will simulate.

Â 11:20

Now I've started from my previous

Â configuration by just reversing the velocity.

Â And you'll see that the system evolve in a very unlikely situation

Â back to its initial state.

Â Okay so this is actually a very sensitive simulation.

Â If you do a very small error, like rounding a truncation in your arithmetic

Â but this is not the case here because we are just doing boolean, and

Â the computer can do boolean arithmetic exactly.

Â But assume that now I do the same experiment but I add just a little error,

Â so it mean that I added just one particle somewhere, okay.

Â And then you see that I don't go back to my initial state.

Â All the particle left in this

Â compartment are those that actually interacted with my extra particle.

Â So it mean that in practice,

Â even though Newton mechanics is a time reversal in variant.

Â There's no chance to produce it because there's no chance to know the system with

Â enough accuracy to change all of it velocity with enough accuracy.

Â If you have any perturbation or

Â error you will lose this property of exact time reversible.

Â But in this fully discrete world you can still play with that.

Â So, now I'd like to show you some other example.

Â Like for instance, this is another lattice gas,

Â which of course illustrates the motion of a sand grain and

Â this mimics the hourglass.

Â So, it's just a tool of what you can do with this type of a model.

Â That's another example where you combine the free dynamics

Â with some grain particle which you have here is in the snow.

Â So the snow falls and piles up and you can have

Â a Christmas card out of your simulation, if you want.

Â You can, of course, do other then free dynamics, for instance, you can

Â ask your particle to have collision that do not conserve momentum, only mass,

Â 13:39

but produce what's called diffusion, as you've seen in one of the chapter.

Â And here's a diffusion with a special case,

Â where you have in the middle, a particle at rest.

Â And each time, when another particle sticks to it,

Â it will stay at rest and form this cluster.

Â Okay, so this is called a DLA.

Â Diffusion limited aggregation, and it's a very common pattern in

Â nature that you can reproduce with this simple system.

Â 14:32

band, where you have a test tube [COUGH] with a gel and some substance.

Â And at the inlet of the test tube,

Â you inject some chemicals, which will diffuse into the test tube.

Â And by diffusing, it precipitate with the substance that was already

Â in the test tube, forming separate bands, and that's what of course was

Â extremely surprising to the chemist who found out at the end of the 1800.

Â Is why is it not a continuous process and

Â he also found there's a nice geometrical rule describing this.

Â And this is an example where I show you can simulate this with and

Â get all this property right, because you put,

Â write mesoscopic approximation of the reaction diffusion process.

Â 15:30

Okay, [COUGH] so I think what we have seen in this lattice

Â class is that you may have wrong behavior from since

Â we saw this anisotropy in the lattice class, and

Â this is due to not enough symmetry in your lattice.

Â For other process like diffusion, that's not a problem.

Â Again it means that if you want to control this type of thing

Â you need to understand that you cannot do everything or anything you want.

Â You should be sure that physic is properly put into your model.

Â For instance in this HPP gas, you can see that yes, you conserve momentum.

Â But actually you conserve too much momentum.

Â You conserve momentum along each of the line of the lattice or each of the column.

Â And that's not very good.

Â You also have this checkerboard invariant which means that everything which with

Â a time key was on a let's say, white cell of a checkerboard.

Â And the next time the step will be on the black one, and so you don't interact with

Â all a particle you actually just partition your system in two sub-system and

Â that's may be not what you want.

Â