# Boltzmann equation and the meaning of the marginals

+ 4 like - 0 dislike
320 views

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?

This post imported from StackExchange MathOverflow at 2016-10-07 22:42 (UTC), posted by SE-user RaphaelB4

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsO$\varnothing$erflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.