We know that Wiener process $\omega(t)$ characterized by the probability distributions:

$$p(\omega_0,t_0) = \frac{1}{\sqrt{4\pi Dt}}\exp\left(-\frac{\omega_0^{2}}{4D t_0}\right)$$
$$p(\omega_1,t_1|\omega_0,t_0) = \frac{1}{\sqrt{4\pi Dt}}\exp\left(-\frac{(\Delta\omega)^2}{4D \Delta t}\right)$$

is not differentiable as the derivative is arbitrarily large. We know that Wiener process is diffusion of a particle in one dimension. What implication does this have for flux equation given by Fick's first law:

$$J = -D\nabla C$$

Does this mean we cannot obtain velocity of of a species by differentiating its concentration profile?

  • $\begingroup$ You say that the process "is not differentiable as the derivative is arbitrarily small". This doesn't seem right to me. A time derivative can not be well-defined because the path of the Wiener process has (almost certainly) unbounded variation in any interval $(t_1,t_2)$. So, wouldn't it be more accurate to say derivatives are infinitely large? See math.stackexchange.com/questions/3258009/… $\endgroup$ – TBissinger Jan 1 at 1:30
  • 1
    $\begingroup$ I stand corrected. I should have said that the probability that the derivative is smaller than an arbitrary constant is zero. $\endgroup$ – Brownian_Motion Jan 1 at 22:18

The two equations stand on different theoretical foundations. Let's start with the context that Fick's law is usually encountered in:

For any density $n(r,t)$ of a conserved variable, the value of $n(r,t)$ integrated over some volume can only change by fluxes through the surface of the volume. This can be expressed in the differential equation $$\partial_t n + \nabla \cdot j = 0.$$ This equation is exact and can even be derived by (say classical) equations of motion when there is some microscopic representation of $n(r,t)$. One interprets $j(r,t)$ as the (vector-valued) flux density corresponding to $n(r,t)$. This is all fair and nice, but this equation is not closed, the description of an unknown $n(r,t)$ is just passed down to some unknown $j(r,t)$. To close it, one can introduce an approximation, usually called a phenomenological relation. That's Fick's law for concentrations or Fourier's law for heat transport and so forth. The statement is that on average, the flux density $\langle j(r,t)\rangle$ associated to a density $\langle n(r,t)\rangle$ should be directed such that it counteracts spatial changes in $\langle n(r,t)\rangle$. The current should thus point from regions of large values of $\langle n(r,t)\rangle$ to regions of low values of $\langle n(r,t)\rangle$, which is just antiparallel to the direction of the gradient. Et voilà, that's Fick's law as you gave it, and it is valid approximately on large enough scales. You need coarse-graining for this to make sense. Without coarse-graining, you can't even speak about a sufficiently smooth function with a well-defined gradient. Once you insert a phenomenological relation like Fick's law, the equations essentially become diffusion equations which you can solve to see how an initially inhomogeneous distribution of $\langle n(r,t)\rangle$ relaxes to a homogeneous one (in the absence of driving forces). Note that there are still arbitrary choices left for the exact averaging procedure denoted by $\langle \cdots \rangle$, typically one integrates all densities over volumes around $r$ that are small compared to the system size but large compared to some interatomic distance or some length scale dictated by the forces involved in the interaction.

The Wiener process works on a different level. To someone modelling the physics phenomenon of Brownian motion, it describes individual tracer particles that are randomly displaced due to collisions with smaller bath particles. It is true, you can not define the velocity of the particle that follows a trajectory according to the Wiener process since the curves are non-rectifiable. But this velocity is also something else than the flux density $j(r,t)$ of the previous description. Especially when you want to compare to Fick's law, because there is no coarse-graining involved when using Brownian motion. So the ill-defined velocity of a single Brownian particle should not be confused with the flux density that is the result of a coarse-graining average over a great many number of particles.

And indeed, these two should not contradict one another. After all, Brownian motion much like the phenomenological approach will predict diffusion of initial density inhomogeneities, as you can see in your relation for $p(\omega_0,t_0)$. They are not mutually exclusive, much rather they give different view points on the phenomena, where the view point surrounding Fick's law is closer to a macroscopic description while Brownian motion yields more inside into the role of spontaneous fluctuations.

  • $\begingroup$ This was super helpful, thank you!! $\endgroup$ – Brownian_Motion Jan 1 at 22:38
  • $\begingroup$ Glad I could help. If you like, you may upvote or accept the answer. $\endgroup$ – TBissinger Jan 5 at 11:10
  • $\begingroup$ I did, it does not show up since I have less than 15 reputation. $\endgroup$ – Brownian_Motion Jan 5 at 16:26

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