The NCMC procedure is performed via a protocol ⇤, which utilizes a sequence of perturbation kernels a t (x, y) and propagation kernels K t (x, y), where x and y are microstates of the system. By "kernels" we mean conditional probability distributions, p, that we can evaluate pointwise and draw samples from. Furthermore, each kernel p must satisfy the requirement that if p(x, y) > 0 then p(y, x) > 0, for all pairs x, y.