How Does the Ehrenfest Wind-Tree Model Describe Particle Dynamics?

[GalileoGalilei]

Homework Statement

A collection of fixed scatterers ('trees') are placed on a plane at random. The trees are oriented squares with diagonals along the x- and y-directions (cf attached picture). The number of trees per unit volume is n, each side is of length a, and na^2 << 1. There are moving particles ('wind') that do not interact with each other, but do collide with the trees. The wind particles can move in four directions, labeled 1,2,3,4. Let F_i(\textbf{r},t) = the number of wind particles at \textbf{r} moving in direction i at time t.

(a) Derive an equation for F_i(\textbf{r},t).
(b) Is there an H-theorem ? (Suppose the system is spatially homogeneous, F_i(\textbf{r},t)=F_i(t), independent of \textbf{r})
(c) Find a solution \left\{ F_i(t)\right\} in terms of \left\{F_i(0)\right\}. What happens if t \rightarrow \infty ? (You will need to diagonalize a 4\times 4 matrix)

Homework Equations

I might need Boltzmann equation for dilute gas.

The Attempt at a Solution

I just began reading a book which is Introduction to Chaos in Nonequilibrium Statistical Mechanics. This exercise is at the end of a chapter on Boltzmann Equation and Boltzmann's H-theorem : I have some diffuclties to know how to begin solving it.

I hope here someone can help me, thanks in advance.

 

[Oxvillian]

Hello GalileoGalilei - looks like an interesting problem!

To get started, I would try to write down an equation for \frac{d}{dt}F_1, the rate of change of the F_1 population. As time goes by, the F_1 population will be depleted because some of it will be scattered (in equal measure) into the 2 and 4 directions. Meanwhile the F_1 population will also be augmented by incoming scattering from the F_2 and F_4 populations.