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.
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.