Microcanonical Langevin Monte Carlo

We propose a method for sampling from an arbitrary distribution $\exp[-S(\x)]$ with an available gradient $\nabla S(\x)$, formulated as an energy-preserving stochastic differential equation (SDE). We derive the Fokker-Planck equation and show that both the deterministic drift and the stochastic diffusion separately preserve the stationary distribution. This implies that the drift-diffusion discretization schemes are bias-free, in contrast to the standard Langevin dynamics. We apply the method to the $\phi^4$ lattice field theory, showing the results agree with the standard sampling methods but with significantly higher efficiency compared to the current state-of-the-art samplers.