Mean-field, ensemble-chain, and adaptive samplers have historically been viewed as distinct approaches to Monte Carlo sampling. In this paper, we present a unifying {two-system} framework that brings all three under one roof. In our approach, an ensemble of particles is split into two interacting subsystems that propose updates for each other in a symmetric, alternating fashion. For the memoryless two-system samplers, this cross-system interaction ensures that the finite ensemble has $ρ^{\otimes 2N}$ as its invariant distribution; for finite-adaptive variants, exact stationarity applies after the adaptation phase is frozen. The two-system construction reveals that ensemble-chain samplers can be interpreted as finite-$N$ approximations to an ideal mean-field sampler; conversely, it provides a principled recipe for discretizing mean-field Langevin dynamics into tractable parallel MCMC algorithms. The framework also connects naturally to adaptive single-chain methods: by replacing particle-based statistics with time-averaged statistics from a single chain, one recovers analogous adaptive dynamics in the long-time limit without requiring a large ensemble. We derive novel two-system versions of both overdamped and underdamped Langevin MCMC samplers within this paradigm. Across synthetic benchmarks and real-world posterior inference tasks, these two-system samplers -- which use a single BCSS-2 integrator step per Metropolis--Hastings accept/reject, in contrast to the long-trajectory style of HMC/NUTS -- exhibit substantial performance gains over No-U-Turn Sampler baselines, achieving higher effective sample sizes per gradient evaluation and markedly higher wall-clock throughput. On higher-dimensional posteriors, the adaptive MAKLA-BCSS-2 methods remain stable and achieve substantially better per-gradient efficiency and wall-clock throughput than the NUTS variants in our benchmark suite.

翻译：均值场、集成链与自适应采样器历来被视为蒙特卡罗采样的不同方法。本文提出统一的{双系统}框架，将三者纳入同一体系。在该方法中，粒子集成被分为两个相互作用的子系统，以对称交替方式为对方提议更新。对于无记忆双系统采样器，这种跨系统交互确保有限集成以其不变分布$\rho^{\otimes 2N}$为目标；对于有限自适应变体，冻结适应阶段后精确平稳性仍然成立。双系统构建揭示，集成链采样器可被解释为理想均值场采样器的有限$N$近似；反之，它提供了将均值场朗之万动力学离散化为可并行MCMC算法的原则性方法。该框架自然地与自适应单链方法关联：通过用单链时间平均统计量替代基于粒子的统计量，可在长时间极限下恢复类似自适应动力学而无需大规模集成。我们在此范式下推导了过阻尼与欠阻尼朗之万MCMC采样器的新型双系统版本。在合成基准与真实后验推理任务中，这些双系统采样器——仅使用单个BCSS-2积分器步长进行Metropolis-Hastings接受/拒绝（与HMC/NUTS的长轨迹风格不同）——相较于No-U-Turn采样器基线表现出显著性能提升：每次梯度评估获得更高有效样本量，并实现明显更高的物理时间吞吐量。在更高维后验分布上，自适应MAKLA-BCSS-2方法保持稳定，且在我们基准测试套件中相比NUTS变体实现了显著更优的每梯度效率与物理时间吞吐量。