= Solution
For a chosen bead, the uniform displacement cube is inversion symmetric, so the <proposal distribution> satisfies $q(X\to Y)=q(Y\to X)$. The target canonical density is $\pi(X)\propto\exp[-\beta U(X)]$ for allowed configurations. With $a(X,Y)=\min[1,e^{-\beta(U(Y)-U(X))}]$,
$$
\boxed{\pi(X)q(X,Y)a(X,Y)=q(X,Y)\min[\pi(X),\pi(Y)]=\pi(Y)q(Y,X)a(Y,X).}
$$
This proves <detailed balance>. Overlapping trials have zero target weight and are rejected. The proposal must retain rejected trials as self-transitions; repeatedly proposing until a nonoverlapping move appears would generally introduce a configuration-dependent proposal normalization and need a Hastings correction. <Detailed balance> establishes stationarity, while sufficient sampling also requires practical exploration of the accessible state space.
Back to article page
