• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,054 questions , 2,207 unanswered
5,345 answers , 22,719 comments
1,470 users with positive rep
818 active unimported users
More ...

  Boltzmann equation and the meaning of the marginals

+ 4 like - 0 dislike

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

asked Sep 28, 2016 in Theoretical Physics by RaphaelB4 (20 points) [ revision history ]
edited Oct 7, 2016 by Dilaton
isn't this the essence of the ergodic hypothesis --- averages over initial conditions are equivalent to temporal or spatial averages?

This post imported from StackExchange MathOverflow at 2016-10-07 22:43 (UTC), posted by SE-user Carlo Beenakker
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 ...

This post imported from StackExchange MathOverflow at 2016-10-07 22:43 (UTC), posted by SE-user Vincent Beffara

Your answer

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):
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:
Then 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).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights