  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. In contrast to the common belief, 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 applied MCLMC on a lattice $\phi^4$ model, where
Hamiltonian Monte Carlo (HMC) is currently the state-of-the-art integrator. For
the same accuracy, MCLMC converges 12 times faster than HMC on an $8\times8$
lattice. On a $64\times64$ lattice, it is already 32 times faster. The trend is
expected to persist to larger lattices, which are of particular interest, for
example, in lattice quantum chromodynamics.
