+ 1$ and where $T^*_i$ is the constraint temperature used for the $i$th $E_\mathrm h$ move, $T^*_{0} = T^*_A$ is some temperature sufficiently small that we can consider the constraints to be exactly satisfied, and $\alpha$ is the rate of increase of the temperature which should be sufficiently small that the adiabatic condition is satisfied. Even though $T^*_A$ and $\alpha$ are small, we can pick $n$ sufficiently large that $T^*_{m+1} = T^*_B = T^*_A
\exp(\alpha m)$ is finite.

In addition, we choose the step size for the $i$th $E_\mathrm h$ move to be $d_i = k \sqrt{T^*_i}$ where $k$ is a constant. In traditional Monte Carlo, we normally pick $k$ to maximize the diffusion rate which at the $i$th step is roughly $$D_i = \frac{\langle (\Gamma_i - \Gamma_{i-1})^2\rangle}2
\sim \frac12 A d_i^2,$$ where $A$ is the mean acceptance rate and $\langle\ldots\rangle$ denotes an ensemble average. Maximizing the diffusion rate usually results in a rather small acceptance rate $A \sim 0.1$ because rare large steps can lead to faster diffusion than frequent small steps. However, in our application, where we want the system to remain in equilibrium as we vary the temperature, rare large steps are [*bad*]{}. So we pick $k$ 