Diffusion in a harmonic potential

Question

I am just looking for the correct Ansatz to a one-dimensional diffusion equation with a harmonic potential.

Let $n(x,t)$ be the density of particles and $\Phi(x)=\frac{1}{2}\alpha x²$ a potential. Then the current density is $$j(x,t)=-D\frac{\partial n(x,t)}{\partial x}-\sigma \left(-\vec\nabla_x \Phi(x) \right)$$ with $D$ as diffusion coefficient and any kind of "conductivity" $\sigma$. With the equation of continuity $$\frac{\partial n(x,t)}{\partial t}+\frac{\partial j(x,t)}{\partial x}=0$$

one gets the inhomogeneous equation $$\frac{\partial n(x,t)}{\partial t}-D \frac{\partial^2 j(x,t)}{\partial x^2}+\kappa=0$$ with a positive constant $\kappa$.

My only question is whether this reasoning is correct? If so, the rest would be solving the equation by standard method.

Answer 1

Your result has something funny about it: $$\frac{\partial n}{\partial t}-D\color{red}{\frac{\partial j}{\partial x} }+\kappa=0 \tag{your result} $$ The $D$ and $\kappa$ both come from the definition of $j$, so you shouldn't have both of these values and the divergence of $\mathbf{j}$ in the result at the same time. Hence, you should be expecting something like $$\frac{\partial n}{\partial t}-D\color{blue}{\frac{\partial^2n}{\partial x^2}}+\kappa=0 \tag{expected result},$$ which has the diffusion term present.

The current density otherwise looks to be of the form of a drift-diffusion equation often seen in semiconductor physics.