Why is diffusion coefficient = 1/2?

[Old Guy]

Homework Statement

I'm working with a 0ne-dimensional random walk and looking at Fick's second law, below. All I've read seems to take D = 1/2 as a given for ordinary diffusion, but where does this come from? Is there a way to derive it?

Homework Equations

$$\frac{\partial }<br /> {{\partial t}}p\left( {x,t} \right) = D\frac{{\partial ^2 }}<br /> {{\partial x^2 }}p\left( {x,t} \right)$$

The Attempt at a Solution

 

[clamtrox]

Essentially D defines your units, so you are allowed to choose any numerical value. I guess people choose 1/2 because then the "usual" solution is a Gaussian with variance t (instead of √(2Dt)).

 

[Old Guy]

Thank you, but can you expand on your answer a bit? Not sure I understand what you mean in your second sentence.

 

[clamtrox]

You know how to solve diffusion equation with Fourier transforms, right? The result is
$p = \frac{1}{\sqrt{4\pi Dt}} \exp(-x^2/4Dt)$
so if you choose D = 1/2, this becomes a normal distribution with mean 0 and variance t.

 

[Old Guy]

Of course, now I get it! Thank you.