1 \section{Double-integral Kramers' theory}
3 The double-integral form of overdamped Kramers' theory may be too
4 complex for analytical predictions of unfolding-force histograms.
5 Rather than testing the entire \sawsim\ simulation (\cref{sec:sawsim}),
6 we will focus on demonstrating that the Kramers' $k(F)$ evaluations
7 are working properly. If the Bell modeled histograms check out, that
8 gives reasonable support for the $k(F) \rightarrow \text{histogram}$
9 portion of the simulation.
11 Looking for analytic solutions to Kramers' $k(F)$, we find that there
12 are not many available in a closed form. However, we do have analytic
13 solutions for unforced $k$ for cusp-like and quartic potentials.
15 \subsection{Cusp-like potentials}
18 \subsection{Quartic potentials}