Diffusion coefficient in diffusion equation and random walk ?

[xxh418]

Hi all:
Now I have a question about the concept of diffusion coefficient in two cases: the diffusion equation (J=DdT/dx) and the random walk (tao^2=6Dt). My quesion is the two D in two equations are the same or different. If they are different, is there any relationship between them?

Best
Xu

 

[Mapes]

Hi xxh418:
The value of D in the two equations is the same under the restrictions used to derive the equations: uniform, isotropic, and constant diffusivity; no correlation between jumps; no other driving force for flux; and dilute concentrations.

 

[xxh418]

Hi Maple
Thank you for your reply. What does the "dilute concentrations" mean? In random walk, there could be only one atom to diffuse in the solute. For the diffusion equation, there will be a concentration gradient. For the mass(atom) diffusion in solid crystals, for example, the Li diffusion in Si nanowire, the concentration of Li could be relative high. In this case, how we judge if it is dilute concentration?

Regards
Xu

 

[Mapes]

What does the "dilute concentrations" mean?

This detail arises from the fact that diffusion is driven by gradients in chemical potential, not by gradients in concentration. However, the difference is negligible for dilute concentrations. Specifically, "dilute" means that the activity coefficient of the diffusing species is (nearly) independent of concentration. For more details, see, for example, Baluffi et al.'s Kinetics of Materials or any good graduate text on diffusion.