Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / Section E / 27 / e

The initial slope near one is the short-observation regime $\mathcal{T}(T)\sim T/2$. Successively longer observations include trajectories that enter progressively longer-lived traps before reaching the absorbing minimum. The four marked shoulders represent successive relaxation timescales, not four distinct equilibrium phases. A plateau appears when the contributing faster modes have largely completed but the next slow component has not yet substantially changed the conditional mean. The ultimate plateau is the true mean first-passage time, approximately $e^{32.3}$ in the plotted time units.

To estimate a rate, locate the rounding timescale at which that mode stops growing with the observation window, using ${\nu }_{l}T\sim 1$; the final completion of a crossover requires several lifetimes. Graphical transitions lie approximately around $\ln T\sim 8$, 23, 29 and 35, giving only order-of-magnitude estimatesThese are roughly $3\times 10^{-4}$, $10^{-10}$, $3\times 10^{-13}$ and $6\times 10^{-16}$, with factors of several uncertainty from reading broad crossovers. Plateau heights are weighted sums $\sum A_l/{\nu }_l$ divided by completed probability, not individually $1/{\nu }_l$; in particular the final plateau at $\ln \mathcal{T}\simeq 32.3$ does not require the slowest lifetime to be $e^{32.3}$. The table fixes barriers and basin connectivity, but supplies no temperature or attempt-frequency convention that would justify exact rates directly from the energies.