1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The Ehrenfest wind-tree model

  1. Jun 15, 2014 #1
    1. The problem statement, all variables and given/known data

    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 [itex]n[/itex], each side is of length [itex]a[/itex], and [itex]na^2 << 1[/itex]. 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 [itex]1,2,3,4[/itex]. Let [itex]F_i(\textbf{r},t) =[/itex] the number of wind particles at [itex]\textbf{r}[/itex] moving in direction [itex]i[/itex] at time [itex]t[/itex].

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

    2. Relevant equations

    I might need Boltzmann equation for dilute gas.

    3. 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.

    Attached Files:

  2. jcsd
  3. Jun 15, 2014 #2
    Hello GalileoGalilei - looks like an interesting problem!

    To get started, I would try to write down an equation for [itex]\frac{d}{dt}F_1[/itex], the rate of change of the [itex]F_1[/itex] population. As time goes by, the [itex]F_1[/itex] population will be depleted because some of it will be scattered (in equal measure) into the 2 and 4 directions. Meanwhile the [itex]F_1[/itex] population will also be augmented by incoming scattering from the [itex]F_2[/itex] and [itex]F_4[/itex] populations.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Threads - Ehrenfest wind tree Date
Classical limit of atomic motion Apr 28, 2017
Time Evolution of Operators Jun 1, 2016
Integrating the solar wind equation May 11, 2016
2D Harmonic Oscillator and Ehrenfest's Theorem Apr 18, 2014