studied.

Static Potential
================

We remind ourselves of the theory of escape times and escape rate from a well of a static potential. These results follow from large deviation and potential theory.

Large Deviation, Potential Theory and Kramers Formula {#chap_sect_kram}
-----------------------------------------------------

Let $V:\mathbb{R}^r:\longrightarrow\mathbb{R}$. Let $x\in\mathbb{R}^r$ be a well and $z_i\in\mathbb{R}^r$ be saddles labelled by $i=1\ldots n$. The saddles would be gateways providing a passage for escape from the well. Define $$\begin{aligned}
\Delta V_i=V(z_i)-V(x)\end{aligned}$$ which is the height difference between the well and the $i$th saddle. For small noise $\epsilon$, an approximate expression can be estimated for the escape time of the particle going through the $i$th saddle. It is well known from the theory of large deviation [@freidlin98], that in the lowest order of the noise $\epsilon$ the mean exit time is given by $$\begin{aligned}
\tau_i=e^{+2\Delta V_i /\epsilon^2}\end{aligned}$$ Inverting this gives the escape rate $$\begin{aligned}
R_i=e^{-2\Delta V_i /\epsilon^2}\end{aligned}$$ and the total escape rate would be to sum over all the saddles $$\begin{aligned}
R=\sum_{i=1}^n R_i
=\sum_{i=1}^n e^{-2\Delta V_i /\epsilon^2}.\end{aligned}$$ The order correction is done by adding a coefficient called Kramers coefficient and the resulting corrected rate is called Kramers rate $$\begin{aligned}
R_i=k_ie^{-2\Delta V_i /\epsilon^2}
\quad
\text{where} 
\quad
k_i=\frac{\sqrt{\left|\nabla^2V(x)\right|}}{2\pi}
\frac{\left|\lambda(z_i)\right|}{\sqrt{\left\Vert \nabla^2V(z_i) \right\Vert}}\end{aligned}$$ 