- #1

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∇

^{2}n

^{*}

for the number density n

^{*}(t,

**r**) = ∫d

^{3}

**v**F

^{*}(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∫Fd

^{3}

**v**

= ∫∂F/∂t d

^{3}

**v**

=∫((∂F/∂t)

_{c}-

**v**.∇F)d

^{3}

**v**

(∂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.∇Fd

^{3}

**v**

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