Fick's Second law from Boltzmann Equation

  • Thread starter The_Foetus
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Homework Statement


Starting from the kinetic equation for the distribution function F*(t, r, v) of some
labelled particle admixture in a gas, derive the self-diffusion equation

∂n*/∂t = D∇2n*
for the number density n*(t,r) = ∫d3vF*(t,r,v) of the labelled particles. Derive also the expression for the self-diffusion coefficient D.

Homework Equations


Boltzmann Equation ∂F/∂t + v.∇F = (∂F/dt)c

The Attempt at a Solution


Dropping the star notation on n and F
∂n/∂t = ∂/∂t∫Fd3v

= ∫∂F/∂t d3v

=∫((∂F/∂t)c - v.∇F)d3v

(∂F/∂t)c term drops out by definition (

As v is a single vector uniform in space, can write it as ∇φ

Therefore ∂n/∂t = ∫∇φ.∇Fd3v

Stuck from here on, don't know how to find D either.
 

Answers and Replies

  • #2
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Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 

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