Diffusion equation in d- dimension

[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.

 

[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.

 

[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. :)