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.

翻译：随机采样算法（如朗之万蒙特卡洛）受热浴中物理系统的启发。其平衡分布为由预设目标分布给出的正则系综，因此必须遵循涨落-耗散定理所规定的涨落与耗散平衡。与普遍认知相反，我们证明涨落-耗散定理并非必需条件，因为只需构型空间分布（而非完整相空间分布）满足正则性。我们提出一种无耗散随机微分方程系统——连续时间微正则朗之万蒙特卡洛（MCLMC）。推导了对应的福克-普朗克方程，并证明其平稳分布为微正则系综，且在构型空间上呈现所需的正则分布。我们证明了MCLMC对任意非零随机性均具有遍历性，且对于光滑凸势函数，期望值呈指数级快速收敛。此外，确定性漂移项与随机扩散项可分别保持平稳分布。这一非常规性质对实际实现颇具吸引力，因为它意味着漂移-扩散离散化方案无偏，仅确定性动力学离散化引入偏差。我们将MCLMC应用于晶格$\phi^4$模型——当前最先进的哈密顿蒙特卡洛（HMC）积分器在该模型上。在相同精度下，对于$8\times8$晶格，MCLMC收敛速度比HMC快12倍；对于$64\times64$晶格，其速度提升已达32倍。该趋势预计在更大晶格（例如晶格量子色动力学特别关注的体系）中持续存在。