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Fick's Second law from Boltzmann Equation

  1. Jan 8, 2015 #1
    1. The problem statement, all variables and given/known data
    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.

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

    3. 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.
  2. jcsd
  3. Jan 13, 2015 #2
    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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