1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Why is diffusion coefficient = 1/2?

  1. Mar 21, 2012 #1
    1. The problem statement, all variables and given/known data
    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?

    2. Relevant equations
    \frac{\partial }
    {{\partial t}}p\left( {x,t} \right) = D\frac{{\partial ^2 }}
    {{\partial x^2 }}p\left( {x,t} \right)

    3. The attempt at a solution
  2. jcsd
  3. Mar 22, 2012 #2
    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)).
    Last edited: Mar 22, 2012
  4. Mar 22, 2012 #3
    Thank you, but can you expand on your answer a bit? Not sure I understand what you mean in your second sentence.
  5. Mar 22, 2012 #4
    You know how to solve diffusion equation with Fourier transforms, right? The result is
    [itex] p = \frac{1}{\sqrt{4\pi Dt}} \exp(-x^2/4Dt) [/itex]
    so if you choose D = 1/2, this becomes a normal distribution with mean 0 and variance t.
  6. Mar 22, 2012 #5
    Of course, now I get it! Thank you.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook