Mario Kasparis

Can we continue with our discussion of mean field limit?

Remember, we consider equations of motion for x i

position and VI, the velocity with linear friction

and some interaction kernel K, which is weighted with 1 over N.

We already introduced the empirical measure, nu N,

which is the sum of the mean of the concentrated measures

at the particle locations and velocities.

And we have the distribution of form, nu N,

like to phi is 1 over N, sum phi of XI, VI.

OK, now we are going to derive an equation, an evolution

equation for nu N.

So we can simply compute a time derivative in this formula.

So since phi does not depend on time explicitly,

we have dT mu N phi is the same as dT mu N phi.

And we can compute this time derivative

here on the right-hand side.

This is 1 over N, sum I from 1 to N.

Then we have a chain rule.

We have first a gradient with respect to XI, the XI dT.

And then we have a gradient with respect to VI of phi scalar

product with dVI dT.

And now, of course, in the next step,

we will insert the equations of motion for dXI dT and dVI dT.

So then we get dT mu N with phi is equal to 1 over N,

sum over I gradient X of phi times VI.

This is the XI dT.

And then we get a gradient of V of phi and XI VI.

And we have to insert dVI dT.

This is minus lambda VI plus 1 over N, sum over J,

K of XI minus XJ.

And now we can use again the empirical measure

to rewrite some of these terms.

At least the first one is very easy.

So let's write this, maybe separate sums.

We have first sum I from 1 to N gradient X phi at XI VI times

VI.

Then we have 1 over N times the sum I from 1 to N gradient

V of phi times minus lambda VI.

And then we have the double sum in the end.

This is then 1 over N squared, sum over I, sum over J,

gradient V of phi times K of XI minus XJ.

So now, if phi is a differentiable function,

then gradient phi times V is still a continuous function.

So we can interpret this in a distributional sense

as mu N applied to this new function, V times gradient X

phi.

This is just 1 over N, sum of the test function,

which is V times gradient phi, evaluated at the location

and velocities of the particles.