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The advection-diffusion equation is given by $$\partial_{t}\rho=-\nabla\cdot\left(\rho\mathbf{v}_{drift}\right)+\nabla\cdot\left(D\nabla\rho\right)\equiv-\nabla\cdot\left(\rho\mathbf{v}_{current}\right).$$ Does this drift velocity $\mathbf{v}_{drift}$ satisfy a Newtonian equation of motion $$m\frac{d}{dt}\mathbf{v}_{drift}=\mathbf{F},$$ where $\mathbf{F}$ is all external, non-diffusion forces?

If so, then should the total time derivative in this equation be expanded using chain rule so that $$\frac{d}{dt}\mathbf{v}_{drift}=\partial_{t}\mathbf{v}_{drift}+\mathbf{v}_{drift}\cdot\nabla\mathbf{v}_{drift}$$ or $$\frac{d}{dt}\mathbf{v}_{drift}=\partial_{t}\mathbf{v}_{drift}+\mathbf{v}_{current}\cdot\nabla\mathbf{v}_{drift}?$$ Which one is correct if either?

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Most usually ${\bf v}_{\rm drift}=- \kappa \nabla V$ as the drift has a similar origin to the electric current in a resistive material. Indeed, as the net current is zero in equilibrium one can use this to derive Einstein relations. For example there is Einstein's famous 1905 relation $$ D= \frac{k_{\rm B}T}{6\pi \eta r} $$ between the diffusion coefficient $D$ for Brownian motion of particles of size $r$ and the viscosity $\eta$ of the fluid in which they move. This allowed Einstein to estimate Avogadro's number.

Alternatively you could be diffusing in a general fluid flow, in which case ${\bf v}_{\rm drift}$ would obey the Navier Stokes equation.

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