# Fick's Second law from Boltzmann Equation

## 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.