Diffusion equation in d- dimension

1. Sep 24, 2008

mhill

i know that idea would seem a bit weird but,

let us suppose we have a surface or volume in d- dimension, here d can be any real number (fractional dimension) the question is that we do not know what value 'd' is

$$\frac{\partial \phi}{\partial t} = D\,\Delta \phi$$

D is a diffusion (constant) coefficient and 'nabla' is the Laplacian in d- dimension.

my question is if we performed a simulation of a 'diffusion process' could we get the value of 'd' , in other words, depending the dimension of space the diffusion process is completely different.

2. Sep 24, 2008

Mapes

The solution for an instantaneous localized source in d-dimensional infinite media is

$$C(r,t)=\frac{N}{(4\pi D t)^{d/2}}\exp({\frac{-r^2}{4Dt}})$$

(from Baluffi's Kinetics of Materials). N is the total amount of material initially at the source, and r is the distance from the source. I've never seen it used for fractional values of d, but the idea is intriguing.

3. Sep 24, 2008

mhill

thanks Mapes, the idea is 'does the dimension of space have physical consequences ? ' for example i was thinking if we could determine what dimension a surface, curve would have using physical methods. :)