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.