论一维 Metropolis 算法的最佳弛豫速率
On the optimal relaxation rate for the Metropolis algorithm in one dimension
AI总结：
研究一维约束势中 Metropolis 算法的弛豫，发现双峰跳跃分布引入振荡的 CDW 机制，构建扩散、拒绝与 CDW 的相图，确定最佳弛豫速率位于三相点。
摘要：
我们研究对应于被困在一维约束势中单个粒子的 Metropolis Monte Carlo 算法的弛豫过程，其偶跳跃分布确保动力学满足细致平衡。先前的工作表明，对于平滑跳跃分布，最快弛豫速率是扩散与拒绝主导动力学竞争的结果。在本研究中，我们证明对于双峰跳跃分布，一种新机制开始起作用，其中弛豫动力学既不由扩散也不由拒绝主导：本征模式采用振荡形式，令人联想到电荷密度波（CDW）——因此我们将此新机制称为 CDW 机制。结合数值与分析技术，我们表征了对应于扩散、拒绝和 CDW 的参数区域及它们之间的转变线——即构建了相图。最佳弛豫速率位于相共存的三相点，即转变线（扩散-拒绝、扩散-CDW 和 CDW-拒绝）的交点。我们的理论框架通过与主方程的数值对角化进行了验证。我们还简要讨论了优化向平衡弛豫速率的更复杂尝试。
英文摘要：
We study the relaxation of the Metropolis Monte Carlo algorithm corresponding to a single particle trapped in a one-dimensional confining potential, with even jump distributions that ensure that the dynamics verifies detailed balance. Previous work suggested that, for smooth jump distributions, the fastest relaxation rate is obtained as a result of the competition between diffusive and rejection-dominated dynamics. In this work, we show that a new regime comes into play for two-peaked jump distributions, where the relaxation dynamics is neither dominated by diffusion nor rejection: the eigenmodes adopt an oscillatory form, reminiscent of charge density waves (CDW) -- thus we term this new regime the CDW regime. Using a combination of numerical and analytical techniques, the parameter regions corresponding to diffusion, rejection, and CDW are characterised, as well as the transition lines between them -- i.e. a phase diagram is built. The optimal relaxation rate is located at the triple point of phase coexistence, where the transition lines (diffusive-rejection, diffusive-CDW, and CDW-rejection) intersect. Our theoretical framework is checked versus the numerical diagonalisation of the master equation. We also briefly discuss more sophisticated attempts at optimising the relaxation rate to equilibrium.
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