approximation only works for small noise. This is because the noise is so small the particle will only escape when the maximum probability to escape has arrived. When the minimum probability to escape is present it will almost never escape. This is the idea behind the perfect phase approximation.

Notice one subtlety behind all the theory presented in this Section. The derivations involved probabilities of escape $p$ and $q$ and the escape rates $R_{-1+1}$ and $R_{+1-1}$. But it was assumed that $p$, $q$, $R_{-1+1}$ and $R_{+1-1}$ are accurately known no matter how large or small the noise level $\epsilon$ is and no matter how fast or slow the driving frequency $\Omega$ is. But such ideal expressions for $p$, $q$, $R_{-1+1}$ and $R_{+1-1}$ are not known.

When we come to do the analysis in Section \[sum\_chap\_results\], the $p_{tot}$ is calculated with the approximation $p_{tot}\approx p_+(t,0)$. When the rates $R_{-1+1}$ and $R_{+1-1}$ are needed they are calculated using Kramers formula as though it is escape from a static potential in the small noise limit. This means an oscillatory potential is being approximated by a static potential which is the adiabatic approximation.

In the paper [@adiabatic_large_deviation_herrmann2006] the adiabatic approximation was justified in the 