I have a question related to the boltzmann equation and the meaning of the marginals.
Let me first introdiuce the model and notation :
(see for example https://arxiv.org/abs/1208.5753)
We study the evolution of a gaz of particles ZN(t)=(X1(t),V1(t),X2(t),V2(t)...XN(t),VN(t)) evolving according to a Hamilton Jacobi equation with a defined hamiltonien H. Because of the indistinguability we focus on
μN(t)=1N∑iδXi(t),Vi(t)
At t=0, the exact initial is not known, but is chosen randomly according to a N particles distributions fN(X1,V1,X2,V2,...).
The probability of finding the particles for t>0, evolving after the initials condition is then given by a fN(ZN,t) which follow the Liouville equation. We then define de first marginal :
f(1)N(X1,V1,t)=∫fN(X1,V1,X2,V2,...,XN,VN)dX2dV2...dXNdVN
Now here is my question : Can we justify the statement : `` f(1)N is the related distribution of the empirical measure μN.'' ? And is it really the object physics use '' ?
Why is it not obvious ? f(1) is a microscopic function average over the initial condition. while the ``mesoscopics density'' of the physicists is for one initial condition the average over a not too small domain.
Let take a example. The domain is a torus and let define fN as follow a point p is chosen randomly uniformly over the domain, then the N particles are put independently in a small sphere around p. In this model because of the symmetry f(1) is constant over all the torus but μN is supported over a small sphere a s.
So the two of them are different. What we know is that if at t=0 particles are completely independent then they are equal because of the strong law of large number but for general dynamics this should not be true for t>0.
Do any one know some rigorous result over the relation between the two objects ? Or an approach of Boltzmann equation using directly the ``mesoscopic function'' instead of the marginals?
This post imported from StackExchange MathOverflow at 2016-10-07 22:42 (UTC), posted by SE-user RaphaelB4