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start this module on microdynamics of lattice gasses.

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

movement of our particle in a gas.

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

some discrete possible velocity.

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

they take another direction.

So this is illustrated in this slide.

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

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

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

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

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

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

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

v i entering the side at the given time.

So now we just put another subscript which is in or

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

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

So with this, you can formulate

the microdynamics of this particle with these two equations.

So first you get a new distribution of out going particle

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

interaction of all the particle at this position.

And then this propagation telling you that If you are out

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

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

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

at the next time slot, okay?

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

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

express in a few moments, okay?

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

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

at [COUGH] every lattice point.

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

it mean that it's actually an incoming particle.

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And if the collision term would be zero,

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

It'd just keep moving straight.

But that wouldn't be a very interesting problem.

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

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

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

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

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

we have four directions.

So I plus two is the opposite direction of I.

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

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

So for this HPP model I have only four possible

velocity which in a Cartesian coordinate are those, and

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

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

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

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

But now suppose it's 1.

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

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

And no other particle in the vertical direction.

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

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

So you can that you do it yourself,

quietly show that this expression is a Boolean expression,

so the quantity can be 0 or 1.

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

slide, okay.

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

in physics we like to talk about density and velocity.

So, what is the density?

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

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

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

and that even site, okay?

And actually you can show that the outgoing mass,

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

is exactly conserved by the rule.

And this is a property of collision terms, so

you can explicitly compute that from this expression by

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

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

Which is by definition the product of the density

by the velocity field of that site.

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

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

So in case of mass equal 1,

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

And that can give me quantity j, but

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

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

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So, I'd like to show some demos and

maybe give you some information about this demo.

And the first one

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

When you simulate that, and

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

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

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

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

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

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

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

in the middle propagates away as a wave.

But it has a very weird shape.

Okay, you will expect a round shape for

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

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

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

you see that things are going much better.

So you still see a slight hexagonal structure.

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

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

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

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

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

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

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

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

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

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

there is no rounding error, no loss of information.

So this is illustrated with this experiment.

So you see, this is my gas HPP, so

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

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

they start to invade the empty right compartment.

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

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

is totally deterministic without any errors, meaning that

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

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

if you reverse the velocity the particles.

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

those going left, I told them to go right.

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

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

Okay, that's the time we will simulate.

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Now I've started from my previous

configuration by just reversing the velocity.

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

back to its initial state.

Okay so this is actually a very sensitive simulation.

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

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

the computer can do boolean arithmetic exactly.

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

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

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

All the particle left in this

compartment are those that actually interacted with my extra particle.

So it mean that in practice,

even though Newton mechanics is a time reversal in variant.

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

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

If you have any perturbation or

error you will lose this property of exact time reversible.

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

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

Like for instance, this is another lattice gas,

which of course illustrates the motion of a sand grain and

this mimics the hourglass.

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

That's another example where you combine the free dynamics

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

So the snow falls and piles up and you can have

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

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

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

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

And here's a diffusion with a special case,

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

And each time, when another particle sticks to it,

it will stay at rest and form this cluster.

Okay, so this is called a DLA.

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

nature that you can reproduce with this simple system.

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

And at the inlet of the test tube,

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

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

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

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

Is why is it not a continuous process and

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

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

get all this property right, because you put,

write mesoscopic approximation of the reaction diffusion process.

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

class is that you may have wrong behavior from since

we saw this anisotropy in the lattice class, and

this is due to not enough symmetry in your lattice.

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

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

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

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

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

But actually you conserve too much momentum.

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

And that's not very good.

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

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

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

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

that's may be not what you want.