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 $Z_{N}(t)=(X_{1}(t),V_{1}(t),X_{2}(t),V_{2}(t)...X_{N}(t),V_{N}(t))$ evolving according to a Hamilton Jacobi equation with a defined hamiltonien $H$. Because of the indistinguability we focus on
$$ \mu_{N}(t)=\frac{1}{N}\sum_{i}\delta_{X_{i}(t),V_{i}(t)} $$
At $t=0$, the exact initial is not known, but is chosen randomly according to a $N$ particles distributions $f_{N}(X_{1},V_{1},X_{2},V_{2},...)$.
The probability of finding the particles for $t>0$, evolving after the initials condition is then given by a $f_{N}(Z_{N},t)$ which follow the Liouville equation. We then define de first marginal :
$$ f_{N}^{(1)}(X_{1},V_{1},t)=\int f_{N}(X_{1},V_{1},X_{2},V_{2},...,X_{N},V_{N})dX_{2}dV_{2}...dX_{N}dV_{N} $$
Now here is my question : Can we justify the statement : `` $f_{N}^{(1)}$ is the related distribution of the empirical measure $\mu_{N}$.'' ? 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 $f_{N}$ 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 $\mu_{N}$ 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?
