A collection of $N$ point particles is held inside a cubic box of side $L$. The walls of the box are sticky. This means that when a particle collides with a wall, it stays attached for a short period of time $\tau$, and then it is released. When it is released, it goes out with the same velocity that it would have if the collision had been elastic in the first place.
What is the spatial distribution of particles? For simplicity, you can assume that only one of the walls is sticky, and the rest are normal. In this case there is only one gradient to be concerned with.
If there were no sticky wall, the distribution would be uniform. I have the intuition that the presence of the sticky wall will increase the concentration of particles near the wall. Think of ergodicity: since the particles spend more time near the wall ($\tau$ extra units), that means that the probability of finding particles near the wall should also be larger. But I am not sure how to prove this, nor how to find the quantitive distribution.
Added: What is the velocity distribution of particles in the bulk of the box? And in the wall?