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In the low-temperature regime, the particle rapidly equilibrates near a minimum and only rarely crosses a neighboring maximum. Applying Laplace's method to the exact current formula gives the Kramers escape ratesEach right or left escape changes position by $2\pi$, henceThe reduced continuous-time random walk on minima has off-diagonal transition ratesand diagonal generator entry $W(l|l)=-(k_{+}+k_{-})$. Its residence time in each well is exponentially distributed with rate $k_{+}+k_{-}$, and the next jump is right with probability $k_{+}/(k_{+}+k_{-})$.

PutOn $[0,2]$, its global maximum is the endpoint $t=2$, whereand the amplitude $\log t$ has value $\log 2$. The endpoint form of Laplace's method therefore givesIndeed, with $u=2-t$, Taylor expansion gives $\varphi (2-u)=2-5u+O(u^2)$ and $\log (2-u)=\log 2+O(u)$; the scale contributing to the integral is $u=O(x^{-1})$. Outside any fixed neighbourhood of $2$, $\varphi$ is bounded above by $2-\delta$, so that part is exponentially smaller. This also controls the integrable logarithmic singularity at zero.

Set $t=zs$. ThenThe fixed phasehas its unique maximum at $s=1$, withThe amplitude at the maximum is $\log z$. The interior-maximum form of Laplace's method givesThusso $\boxed{a=2\pi }$. This is the Laplace asymptotic for the derivative of the Gamma function.