随机格点系统的粗粒化方案与后验误差估计
Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems
AI总结：
针对随机格点系统，提出基于簇展开的粗粒化方案，通过相对熵进行先验和后验误差分析，实现临界行为与磁滞的精确预测。
摘要：
本研究的主要目标是开发随机多体微观模型的粗粒化方案，并通过先验和后验误差分析量化其有效性。本文聚焦于平衡态下相互作用的随机格点系统，例如Ising型模型。所提出的算法源于初始粗粒化近似，该近似可直接通过蒙特卡洛模拟计算，并使用精确与近似粗粒化平衡测度之间的特定相对熵来计算相应的数值误差。随后，我们围绕这个最初且通常不充分的近似进行簇展开，获得更精确的粗粒化方案。簇展开还给出了粗粒化近似的尖锐后验误差估计，可用于构建自适应粗粒化方法。我们给出了一系列数值例子，证明本文开发的粗粒化方案能够准确预测具有中程和长程相互作用的系统中的临界行为和磁滞。我们还展示了这些方案在短程相互作用系统中显著改进了早期粗粒化方案的预测结果。
英文摘要：
The primary objective of this work is to develop coarse-graining schemes for stochastic many-body microscopic models and quantify their effectiveness in terms of a priori and a posteriori error analysis. In this paper we focus on stochastic lattice systems of interacting particles at equilibrium. %such as Ising-type models. The proposed algorithms are derived from an initial coarse-grained approximation that is directly computable by Monte Carlo simulations, and the corresponding numerical error is calculated using the specific relative entropy between the exact and approximate coarse-grained equilibrium measures. Subsequently we carry out a cluster expansion around this first-and often inadequate-approximation and obtain more accurate coarse-graining schemes. The cluster expansions yield also sharp a posteriori error estimates for the coarse-grained approximations that can be used for the construction of adaptive coarse-graining methods. We present a number of numerical examples that demonstrate that the coarse-graining schemes developed here allow for accurate predictions of critical behavior and hysteresis in systems with intermediate and long-range interactions. We also present examples where they substantially improve predictions of earlier coarse-graining schemes for short-range interactions.
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