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I am writing code that simulates a bath of particles. For various reasons, I need to know the answer to the following question in order to implement something for my program.

Suppose we have an (isotropic) bath of particles with number density [tex] n [/tex]. The velocity distribution of the particles is [tex] f(\mathbf{v}) = f(v) [/tex], normalized such that

[tex] n = \int_{0}^{\infty} f(\mathbf{v}) \, d^{3}v. [/tex]

So the mean speed is

[tex] \langle v \rangle = \frac{1}{n} \int_{0}^{\infty} v f(\mathbf{v}) \, d^{3}v. [/tex]

What is the rate (flux) at which particles pass through the boundary of a sphere of radius [tex] R [/tex] centered at the origin? (I want the flux of particles into the sphere, I don't care about the ones that are already inside.)

I also want to include all collisions of particles with the spherical boundary, from those that are just tangential with the boundary to those that pass directly into the sphere. If the calculation has no analytic answer, just formulating the integral would be fine with me so I can do it numerically.

The question seems like a simple ideal gas problem, but I did not find similar questions in the statistical mechanics textbooks that I looked at. And I'm not entirely sure how to do it.

Thanks very much!

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# Flux of particles through a sphere?

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