# Drift velocity in advection-diffusion equation

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?

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.