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In class we are talking about the Maxwell Boltzmann Distribution Equation and the professor said that $d^3p = 4 \pi p^2 dp$.

Im not sure why it doesn't depend on $\phi$ or $\theta$? Is the $d^3p$ the same as volume? Why are we integrating with respect to a sphere and not a cylinder etc.?

Zade Johnston
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Astronomical
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    In general$$\iiint_{\Bbb R^3}f(\vec{p})d^3\vec{p}=\iiint_{[0,,\infty)\times[0,,\pi]\times[0,,2\pi]}f(\vec{p})p^2dp\sin\theta d\theta d\phi,$$but if $f$ is spherically symmetric that's $\int_0^\infty4\pi fp^2dp$. – J.G. Nov 02 '22 at 20:57
  • See this https://physics.stackexchange.com/q/99331/226902 – Quillo Jan 17 '23 at 00:25

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All the possible values of the vector $\vec p$ with magnitudes between $p_1$ and $p_1+dp_1$ can be represented by a sheaf of arrows with their tails at the origin and their heads contained in a spherical shell of inner radius $p_1$ and outer radius $p_1+dp_1$. The volume in $p$-space representing all the vectors having magnitudes between these values is therefore the 'volume' of the spherical shell, that is $4\pi p_1^2dp_1$, or, dropping the subscript because it has served its purpose, $4\pi p^2dp$.

Philip Wood
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  • when you say that their tails are at the origin do you mean 0? So the arrows point from the origin at (0,0) to both the inner shell p and the outer shell p+dp – Astronomical Nov 02 '22 at 18:20
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    Yes, [but I have to add more letters before the comment will load!] – Philip Wood Nov 02 '22 at 18:22
  • so for any sort of distribution will it always be a spherical shell? – Astronomical Nov 02 '22 at 18:29
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    Yes, for any isotropic distribution. – Philip Wood Nov 02 '22 at 18:29
  • I think thats where im having some confusion because Im not sure what really constitutes an isotropic distribution. I had a homework problem that talked about the fraction of hydrogen gas escaping earths gravitational field and $4\pi p^2dp$ is used there but im not sure why. – Astronomical Nov 02 '22 at 18:36
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    Isotropic means no direction favoured. All directions for $\vec p$ are equally probable. – Philip Wood Nov 02 '22 at 18:38
  • are these curves what is meant by a locus curve? – Astronomical Nov 02 '22 at 18:46
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    By "these curves" do you mean the spherical surfaces? I wouldn't call them "locus curves". A locus is a path traced by a point, real or imaginary, usually as the value of some parameter is changed. But I'm no expert on mathematical nomenclature. – Philip Wood Nov 02 '22 at 18:52