has a uniform invariant distribution. A typical zero move samples a new configuration from a distribution which satisfies the symmetry requirement $p(\Gamma';\Gamma) =
p(\Gamma;\Gamma')$, where $p(\Gamma';\Gamma)$ is the probability density of picking a new configuration of $\Gamma'$ given a starting configuration of $\Gamma$. Clearly a sequence of $n$ $E$ moves is itself an $E$ move. From the central limit theorem, a sequence of $n$ zero moves is equivalent, in the limit of large $n$, to selecting the new configuration from a multi-dimensional Gaussian.

Instead of carrying out the $n$ moves with a given energy $E(\Gamma)$, we can consider the case where the energy is given by $E_\lambda(\Gamma)$ which depends continuously on the parameter $\lambda$. A sequence of $n$ $E_\lambda$ moves where $\lambda$ is varied [*adiabatically*]{} in such a way that its initial and final values are $\lambda_0$ is an $E_{\lambda_0}$ move. This follows because adiabatically varied systems are always in equilibrium with the instantaneous value of $\lambda$ [@landau69 §11]. Each $E_\lambda$ move is carried out at a [*fixed*]{} $\lambda$ and $\lambda$ is varied between the moves. In order to satisfy the adiabatic condition, we will need to take $n$ large.

A move from $\Gamma$ to $\Gamma'$ may be subjected to 