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# Fluctuation without dissipation: Microcanonical Langevin Monte Carlo

*Proceedings of the 6th Symposium on Advances in Approximate Bayesian Inference*, PMLR 253:111-126, 2024.

#### Abstract

Stochastic sampling algorithms such as Langevin Monte Carlo are inspired by physical systems in a heat bath. Their equilibrium distribution is the canonical ensemble given by a prescribed target distribution, so they must balance fluctuation and dissipation as dictated by the fluctuation-dissipation theorem. We show that the fluctuation-dissipation theorem is not required because only the configuration space distribution, and not the full phase space distribution, needs to be canonical. We propose a continuous-time Microcanonical Langevin Monte Carlo (MCLMC) as a dissipation-free system of stochastic differential equations (SDE). We derive the corresponding Fokker-Planck equation and show that the stationary distribution is the microcanonical ensemble with the desired canonical distribution on configuration space. We prove that MCLMC is ergodic for any nonzero amount of stochasticity, and for smooth, convex potentials, the expectation values converge exponentially fast. Furthermore, the deterministic drift and the stochastic diffusion separately preserve the stationary distribution. This uncommon property is attractive for practical implementations as it implies that the drift-diffusion discretization schemes are bias-free, so the only source of bias is the discretization of the deterministic dynamics. We apply MCLMC to a $\phi^4$ model on a 2d lattice, where Hamiltonian Monte Carlo (HMC) is currently the state-of-the-art integrator. MCLMC converges 12 to 32 times faster than HMC on an $8\times8$ to $64\times64$ lattice, and we expect even higher improvements for larger lattice sizes, such as in large scale lattice quantum chromodynamics.