Flux of particles through a sphere?

In summary, the rate at which particles pass through the boundary of a sphere in an isotropic bath can be calculated using the flux equation, taking into account the velocity distribution and collisions with the spherical boundary. This integral can be evaluated numerically using a Monte Carlo simulation.
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Hi all,

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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  • #2




Thank you for your question! The rate at which particles pass through the boundary of a sphere can be calculated using the flux equation:

\Phi = \int_{0}^{\infty} \int_{0}^{2\pi} \int_{0}^{\pi} v f(v) \cos(\theta) \sin(\theta) d\phi d\theta dv,

where \Phi is the flux, v is the speed of the particles, \theta is the angle between the particle's velocity vector and the normal vector of the sphere's surface, and \phi is the azimuthal angle. This integral takes into account all possible directions of the particles' velocities and their collisions with the spherical boundary.

To calculate this integral numerically, you can use a Monte Carlo simulation, where you randomly generate particles with velocities following the given velocity distribution and determine how many of them pass through the spherical boundary. You can repeat this process multiple times and take the average to get a more accurate result.

I hope this helps and good luck with your simulation!
 

What is the definition of "flux of particles through a sphere"?

The flux of particles through a sphere is a measure of the number of particles passing through the surface of a sphere per unit time.

What factors affect the flux of particles through a sphere?

The flux of particles through a sphere is affected by the size and shape of the sphere, the concentration of particles, the velocity of the particles, and the properties of the particles and the medium they are passing through.

How is the flux of particles through a sphere calculated?

The flux of particles through a sphere can be calculated using the formula: flux = concentration * velocity * surface area of the sphere.

What is the significance of studying the flux of particles through a sphere?

Studying the flux of particles through a sphere is important in various scientific fields such as physics, chemistry, and biology. It can help us understand the movement of particles in different environments and provide insights into processes such as diffusion and filtration.

Can the flux of particles through a sphere be influenced or controlled?

Yes, the flux of particles through a sphere can be influenced and controlled by changing the factors that affect it, such as the size and shape of the sphere, the concentration and velocity of the particles, and the properties of the particles and medium.

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