An $O(\log N)$ Monte Carlo method for periodic Coulomb systems

TL;DR

DMK-MC addresses the bottleneck of long-range Coulomb energy updates in 3D-periodic systems by adapting the DMK framework to single-particle Metropolis moves. It decomposes the Coulomb kernel into a global smooth part, a hierarchy of localized difference kernels, and a short-range residual, enabling $O(1)$ work per level and overall $O( abla N)$ per trial move for fixed accuracy. The method relies on an adaptive octree and stored incoming plane-wave fields to realize fast energy differences and incremental updates, achieving substantial speedups over recent FMM-based MC approaches while maintaining controlled accuracy. The results on uniform, nonuniform, electrolyte, and colloidal systems demonstrate consistent accuracy and significant practical performance gains, with open-source implementations available for reproducible benchmarking.

Abstract

Efficient Monte Carlo (MC) sampling of many-body systems with long-range electrostatics is often limited by the cost of per-move energy-difference evaluation under periodic boundary conditions. We present DMK-MC, an accelerated MC method that adapts the dual-space multilevel kernel-splitting (DMK) framework to single-particle Metropolis updates. DMK-MC computes the energy change and, upon acceptance, updates the stored incoming plane-wave fields with $O(1)$ work per tree level, yielding an overall $O(\log N)$ expected work per trial move for fixed accuracy. The method decomposes the Coulomb kernel into three components: a global, periodized smooth part; a multilevel sequence of smooth difference kernels whose interactions are restricted to same-level colleague boxes; and a singular residual kernel whose short-range interactions are evaluated directly. Benchmarks on uniform, highly nonuniform, and implicit-solvent electrolyte and colloidal configurations show that DMK-MC consistently outperforms a recent FMM-based $O(\log N)$ Monte Carlo method, delivering several-fold speedups at comparable tolerances.

Figures (4)

• Figure 1: Accuracy and time complexity of DMK-MC for energy-difference evaluation. (a) and (c): relative error and wall-clock time per proposal for systems with uniform distribution. (b) and (d): relative error and time per proposal for systems with highly nonuniform distribution. Dotted lines in panels (c) and (d) show least-squares fits of the form $y = a \log(x) + b$.
• Figure 2: Timing comparison between DMK-MC and the FMM-based method Saunders2021New (log-$x$ scale), showing the per-move wall-clock time for energy-difference evaluation ("propose") and for accepted-move updates ("accept"). The $y$-axis reports time in milliseconds. (a) Uniform distribution. (b) Highly nonuniform distribution.
• Figure 3: (a) Na-Na and Na-Cl RDFs for the 1:1 NaCl electrolyte. Data are shown for DMK-MC and PME-MD at the same prescribed accuracy $\varepsilon=10^{-3}$. (b) Probability profile (P) of the relative error in the acceptance probability $\mathscr{P}_{\mathrm{accept}}$ from DMK-MC at $\varepsilon=10^{-3}$, measured against a higher-accuracy reference solution. The average relative error in $\mathscr{P}_{\mathrm{accept}}$ is $6.57\times 10^{-4}$.
• Figure 4: (a) Colloid--Na and colloid--Cl RDFs for a strongly inhomogeneous colloidal-ion system, and (b) probability profiles of the relative error in the acceptance probability $\mathscr{P}_{\mathrm{accept}}$ from DMK-MC at $\varepsilon=10^{-3}$, measured against a higher accuracy reference solution. The average relative error in $\mathscr{P}_{\mathrm{accept}}$ is $1.47\times 10^{-3}$.

Theorems & Definitions (3)

• Remark 4.1
• Remark 4.2
• Remark 4.3