to maximize $A D_i$ and this will usually result in $A \sim 0.5$. Note that for a given $k$, we have $$D_i \sim C T^*_i,$$ where $C$ is constant provided that the step size is not too large. The overall diffusion can be estimated by summing over the $n$ steps, $$D =  \frac{\langle (\Gamma_n - \Gamma_0)^2\rangle}2 =
\sum_{i=1}^n D_i \sim 2 C T^*_B/\alpha,$$ where we have assumed that successive steps are uncorrelated and we have taken $\alpha\ll 1$ and $T^*_B \gg T^*_A$. We should select parameters, $\alpha$ and $T^*_B$, in order to adjust $D$ so that the $E_\mathrm s$ acceptance rate is $O(1)$.

This method includes internal diagnostics to verify that $\alpha$ is small enough. We define $\overline{\vphantom{A}\ldots}\mathord{\uparrow}$ (resp. $\overline{\vphantom{A}\ldots}\mathord{\downarrow}$) as the average of a quantity over the steps where $T^*_i$ is increasing, i.e., $i \le m+1$ (resp. decreasing, i.e., $i> m+1$). We monitor $\overline{E_\mathrm h(\Gamma_i)/T^*_i}\mathord{\uparrow}$ and $\overline{E_\mathrm h(\Gamma_i)/T^*_i}\mathord{\downarrow}$ and demand that both should be close to the equilibrium value of $N/2$ (where $N$ is the number of hard degrees of freedom). If $\alpha$ is too large, then we would find $$\begin{aligned}
\overline{ E_\mathrm h(\Gamma_i)/T^*_i }\mathord{\uparrow} &\ll& N/2, \\
\overline{ E_\mathrm h(\Gamma_i)/T^*_i }\mathord{\downarrow} &\gg& N/2.\end{aligned}$$ In particular, if the final 