# I What is the propagation speed of a diffusion

#### SirVuk

Hello everybody!

For my water in nanoscaled-pores simulations with SPH I need a value for the characteristic velocity.
My planned approach is to estimate this value by attaining the propagation speed of a diffusion wave.
But I have problems with understanding this process since I find some sources talking about an infinite speed of sound as reference for the propagation speed, while others say that it is a finite value.

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#### Orodruin

Staff Emeritus
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There is no such thing as a characteristic speed for diffusion if you do not somehow set a length or time scale. The reason for this is that diffusion is not a wave. It is described by the diffusion equation $\partial_t u - D\nabla^2 u = \kappa$, where $\kappa$ represents a source term and $D$ is the diffusivity, which has units of length^2/time.

#### SirVuk

There is no such thing as a characteristic speed for diffusion if you do not somehow set a length or time scale. The reason for this is that diffusion is not a wave. It is described by the diffusion equation $\partial_t u - D\nabla^2 u = \kappa$, where $\kappa$ represents a source term and $D$ is the diffusivity, which has units of length^2/time.
ok thanks

but is there a general equation explaining with which speed a diffusion front propagates?

#### mfb

Mentor
There is no well-defined front, but if you look at a fixed concentration, you can determine where this is reached when. You'll find that its propagation slows down over time, so there is no characteristic speed either.

#### Henryk

Gold Member
There is no such thing as 'diffusional wave'. Diffusion is the result of Brownian motion.
However, there is a formula to that is quite useful in estimating the range of the diffusion. Essentially, it is a solution of the diffusion equation; Gaussian function.
The (squared) width of the Gaussian is given by $\sigma ^2 = 2 Dt$
It is quite useful in estimating the effects of the diffusion given the time and the dimensional scales.

"What is the propagation speed of a diffusion"

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