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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?

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  • $\begingroup$ isn't this the essence of the ergodic hypothesis --- averages over initial conditions are equivalent to temporal or spatial averages? $\endgroup$
    – Carlo Beenakker
    Sep 28, 2016 at 12:26
  • $\begingroup$ The effective independence for positive time is usually referred to as "propagation of chaos" (and obtaining it for a large enough time interval is the main contribution of the paper you link to). There are a few methods to prove such a result, but it is never trivial ... $\endgroup$
    – Vincent Beffara
    Sep 28, 2016 at 19:30
  • $\begingroup$ This question possibly needs to be migrated to Physics SE. $\endgroup$
    – user178594
    Apr 2, 2023 at 5:33

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