An efficient algorithm for numerical computations of continuous densities of states
An efficient algorithm for numerical computations of continuous densities of states
复制标题
连续态密度数值计算的有效算法
DOI：
10.1140/epjc/s10052-016-4142-5
复制
发表时间：
2015
期刊：
影响因子：
--
通讯作者：
A. Rago
中科院分区：
文献类型：
--
作者：
K. Langfeld;B. Lucini;R. Pellegrini;A. Rago
在Wang-Landau类型的算法中，蒙特卡罗更新是关于状态密度的，在模拟过程中迭代地进行改进。然后通过标准积分得到配分函数和热力学观测值。在这项工作中，我们最近在这类中引入的方法(LLR方法)被分析和进一步发展。我们的方法是一种无直方图的方法，特别适用于具有连续自由度的系统，它产生了连续的态密度，这在格点规范理论和一些统计力学系统中很常见。我们证明了该方法具有指数误差抑制，允许我们以几乎恒定的相对精度来估计几个数量级上的态密度。我们解释了如何避免遍历性问题，以及如何在这个框架内获得任意可观测的期望值。然后，我们以紧致U(1)格点规范理论为例演示了该方法。对结果的算法参数依赖性进行了深入的研究，并与分析预期的行为进行了比较。我们得到了相变临界耦合和比热峰值的高精确值，晶格尺寸范围从84\Documentclass[12pt]{Minimum}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{matrsfs}\usepackage{upgreek}\setLong{oddsidemarin}{-69pt}\Begin{Document}$8^4$\end{DocumentClass[12pt]{Minimum}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$$20^4$\end{Document}。我们的结果与文献中报道的参考值完全一致，它涵盖了晶格尺寸高达184\Docentclass[12pt]{Minimum}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsFonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{matrsfs}\usepackage{upgreek}\setlong{\oddsidemarin}{-69pt}\Begin{Document}$18^4$\end{Document}。204\Documentclass[12pt]{Minimum}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsfonts}\usepackage{amsbsy}\usepackage{mathsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$20^4$$\end{Document}卷的可靠结果是首次获得的。后一种研究由于在系统的伪临界耦合处形成了很强的亚稳态，到目前为止，即使在具有重要采样方法的超级计算机上也是遥不可及的，已经以有限的计算资源执行了高精度的研究。这显示了该方法在一阶相变研究中的潜力。文中还指出了该方法优于重要性抽样技术的其他情况。
In Wang–Landau type algorithms, Monte-Carlo updates are performed with respect to the density of states, which is iteratively refined during simulations. The partition function and thermodynamic observables are then obtained by standard integration. In this work, our recently introduced method in this class (the LLR approach) is analysed and further developed. Our approach is a histogram free method particularly suited for systems with continuous degrees of freedom giving rise to a continuum density of states, as it is commonly found in lattice gauge theories and in some statistical mechanics systems. We show that the method possesses an exponential error suppression that allows us to estimate the density of states over several orders of magnitude with nearly constant relative precision. We explain how ergodicity issues can be avoided and how expectation values of arbitrary observables can be obtained within this framework. We then demonstrate the method using compact U(1) lattice gauge theory as a show case. A thorough study of the algorithm parameter dependence of the results is performed and compared with the analytically expected behaviour. We obtain high precision values for the critical coupling for the phase transition and for the peak value of the specific heat for lattice sizes ranging from 84\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$8^4$$\end{document} to 204\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$20^4$$\end{document}. Our results perfectly agree with the reference values reported in the literature, which covers lattice sizes up to 184\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$18^4$$\end{document}. Robust results for the 204\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$20^4$$\end{document} volume are obtained for the first time. This latter investigation, which, due to strong metastabilities developed at the pseudo-critical coupling of the system, so far has been out of reach even on supercomputers with importance sampling approaches, has been performed to high accuracy with modest computational resources. This shows the potential of the method for studies of first order phase transitions. Other situations where the method is expected to be superior to importance sampling techniques are pointed out.