Fick's Second law from Boltzmann Equation

[The_Foetus]

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.

 

[anathema]

Hello, thank you for your post. I will provide a possible solution to the problem, but please note that there may be other approaches and solutions that are equally valid.

To derive the self-diffusion equation, we start with the Boltzmann equation:

∂F/∂t + v.∇F = (∂F/∂t)c

We can rewrite this equation in terms of the number density n(t,r) as follows:

∂n/∂t + ∫vd3v∇F = (∂n/∂t)c

Next, we use the definition of the number density n as the integral of the distribution function F over all velocities:

n = ∫Fd3v

Therefore, we can rewrite the above equation as:

∂n/∂t + ∫vd3v∇F = ∫∂F/∂td3v

Now, let's consider the left-hand side of the equation. We can use the chain rule to rewrite the integral over velocity as an integral over position and velocity:

∫vd3v∇F = ∫d3r∫vd3v(∇F)

= ∫d3r∫vd3v(∂F/∂t + v.∇F)

= ∫d3r(∂F/∂t + ∫vd3vv.∇F)

= ∫d3r(∂F/∂t + ∇.∫vd3vvF)

= ∫d3r(∂F/∂t + ∇.(vF))

= ∫d3r(∂F/∂t + v.∇F + F.∇v)

= ∫d3r(∂F/∂t + v.∇F + F.(∂v/∂r))

= ∫d3r(∂F/∂t + v.∇F + F.(∇v))

= ∫d3r(∂F/∂t + v.∇F + F.∇(∇φ))

= ∫d3r(∂F/∂t + v.∇F + F.∇2φ)

= ∫d3r(∂F/∂