\section{Double-integral Kramers' theory}

The double-integral form of overdamped Kramers' theory may be too
complex for analytical predictions of unfolding-force histograms.
Rather than testing the entire sawsim simulation, we will focus on
demonstrating that the Kramers' $k(F)$ evaluations are working
properly.  If the Bell modeled histograms check out, that gives
reasonable support for the $k(F) \rightarrow \text{histogram}$ portion
of the simulation.

Looking for analytic solutions to Kramers' $k(F)$, we find that there
are not many floating around in a finished form.  However, we do have
analytic solutions for unforced $k$ for cusp-like and quartic
potentials.

\subsection{Cusp-like potentials}


\subsection{Quartic potentials}

