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Thimble
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Homework Statement
A two-dimensional circular region of radius a has a gas of particles with uniform
density all traveling at the same speed but with random directions. The wall of the
chamber is suddenly taken away and the probability density of the gas cloud subsequently
satisfies
$$ \rho(r,\theta;t)=\frac{1}{\pi a^2}\int_0^{2\pi} \frac{d\phi}{2\pi} \Theta[a^2-r^2-(vt)^2+2rvtcos(\theta-\phi)]$$
where (r, θ) are the polar-coordinates of the point at issue, t is the time, v is the velocity
of the gas,Φ is the random orientation and Θ[x] vanishes if x is negative and is unity
if x is positive. Employing the equations below show that for each Φ and t the region mapped out by Θ is a circle and hence that the density is normalised.
Homework Equations
$$x=rcos(\theta)$$
$$y=rsin(\theta)$$
$$d_x=vtcos(\phi)$$
$$d_y=vtsin(\phi)$$
The Attempt at a Solution
Firstly I rearranged the theta expression to get
$$\Theta[a^2-(x-d_x)^2-(y-d_y)^2]$$ which I think satisfies the first part of the question with each Φ and t mapping out a circle with radius a and centre (dx,dy)
Now for the second part I'm thinking that I've got to show that the following statement is true
$$\int_{\phi=0}^{2\pi} \int_{r=r_{min}}^{r_{max}} \rho(r,\theta;t)rdrd\theta =1$$
where rmaxand rmin
are the maximum and minimum radii of the region the gas is in at that time which I think is a circle for a>vt and a doughnut shape for a<vt by thinking of the entire region as a series of circles of the same radius with centers moving away from the origin at the same speed in every possible direction.
The stumbling block I'm having is trying to integrate this theta function. My attempt at it was just treating the theta function as one as for all Φ there are points where the expression inside the theta is greater than zero and thus Θ=1, but this gives a constant probability density which doesn't make sense to me as the area over which the gas is in is expanding so the probability density must surely decrease. I've never come across this type of function before and all the examples I've seen on the internet for them all seem to be in one variable rather than 4 (r,θ,t and Φ)so could anyone point me towards any guidance on how to integrate these types of functions? Thanks in advance for any help.
EDIT: Changed upper limit in the first integral to 2π.
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