The gas of hard spheres is a model for a gas in a container, where each particle is a sphere of radius $\epsilon$. The spheres interact with each other and with the container with elastic collisions. The rigorous understanding of this model is far from complete. For example, Boltzmann ergodic hypothesis (i.e., whether this system is ergodic) is a major open problem in ergodic theory. Also, whether the Boltzmann equation can be derived as a limit of this model (aka the Boltzmann-Grad limit) is also open. My question is if the following (more modest) result is known:
Informal statement: Suppose the container is rectangular and initially the whole gas is in half of the container, restricted via a wall. At $t=0$, this wall is removed. Is it true that after sufficiently enough time, the left and right halves will be balanced?
For simplicity, I state it formally for 2d. If one is aware of a 3d result, please state it.
Formal statement: The container is $[-a,a]\times [-b,b]$. The number of particles is $n$. We are interested in the regime where $n$ is large, the radius $\epsilon$ is small and $n\epsilon =\Theta(1)$. This is the Boltzmann-Grad scaling and is chosen this way to ensure that the average distance covered by a particle between two successive collisions is of order 1. The space of the particles' centers is the set $\mathcal{X}$ containing all tuples $(x_1,\dots, x_n)$ with $x_i\in [-a+\epsilon,a-\epsilon]\times [-b+\epsilon,b-\epsilon]$ and with pairwise distances satifying $\|x_i-x_j\|\ge \epsilon$ for $i\neq j$. Let $\mathcal{X_0}$ be the same as $\mathcal{X}$, but with $x_i\in [-a+\epsilon,-\epsilon]\times [-b+\epsilon,b-\epsilon]$. We fix a constant $K$, which will be the average kinetic energy. Now, we generate the initial positions and velocities as follows: we sample $(x_1,\dots,x_n)$ uniformly at random from $\mathcal{X_0}$, and the tuple of velocities $(v_1,\dots,v_n)$ uniformly at random from the sphere in $\mathbb{R}^{2n}$ centered at the origin and with radius $\sqrt{2Kn}$. The task is to prove that if $T$ is a moment in time (independent of the initialization), then with high probability (over the random initialization), at time $T$ the number $n_1$ of particles in the left half, and the number $n_2$ of particles in the right half satisfy $\frac{|n_1-n_2|}{n} \le f(T,N)$, where $f$ goes to zero as $T,N\to \infty$.
