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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.
     
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