in the case of homgeneous temperature $T$,drag coefficient $\gamma$ and conservative force $F(r)=-\nabla V(r)$, the well known Smoluchowski equation is \begin{align} \frac{\partial P(r,t)}{\partial t}&=\frac{1}{\gamma}\nabla\cdot\big[P(r,t)\nabla V(r)+kT\nabla P(r,t)\big]. \end{align} Usually can be written in a more condensed form \begin{align} \frac{\partial P(r,t)}{\partial t}&=\mathcal{L}P(r,t), \end{align} where $\mathcal{L}$ is the corresponding evolution opertaor.

If one writes the Smoluchoski equation in this way, what is the proper way to define the evolution operator?

I simple wrote it as

\begin{align}\label{Eq:Smoluchowski} \mathcal{L}&=\frac{1}{\gamma}\nabla\cdot\big[\nabla V(r)+kT\nabla\big], \end{align}

yet I was told by a peer reviewer that my notation is ambiguous and can be misinterpreted as

\begin{align} \mathcal{L}P(r,t)=\frac{1}{\gamma}\nabla\cdot\big[\nabla (V(r)P(r,t))+kT\nabla P(r,t)\big], \end{align}

In my current work, I refer back to $\mathcal{L}$ and also use it to operate over other variables aside $P(r,t)$ multiple times, thus I need to clarify its definition and avoid any ambiguity.


1 Answer 1


One way to make the order of multiplication/differentiation more clear is to distribute the divergence. I'll use $\Delta=\nabla\cdot\nabla$ to denote the Laplacian. $$ \mathcal{L}P=\frac{1}{\gamma}\nabla\cdot\left(P\nabla V+kT\nabla P\right) \\ =\frac{1}{\gamma}\left[(\Delta V)P+(\nabla V)\cdot\nabla P+kT\Delta P\right] \\ =\frac{1}{\gamma}\left[(\Delta V)+(\nabla V)\cdot\nabla+kT\Delta\right]P \\ $$ And thus we can write $\mathcal{L}=\frac{1}{\gamma}[(\Delta V)+(\nabla V)\cdot\nabla+kT\Delta]$.

That said, there's nothing wrong with defining $\mathcal{L}$ by defining how it acts on functions, i.e. using the expression $\mathcal{L}P=\frac{1}{\gamma}\nabla\cdot\left(P\nabla V+kT\nabla P\right)$ as the defintion of $\mathcal{L}$.


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