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The maximum entropy method in statistical physics: An alternative approach to the theory of simple fluids.

机译：统计物理学中的最大熵方法：简单流体理论的另一种方法。

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摘要

Theories of physics are always developed based on the reasoning method of “deduction”. The studies of entropy from thermodynamics to information theory, however, show a natural reasoning method, “inference”, to be more appropriate in the field of statistical physics.; Making statistical predictions in statistical physics requires solving three problems: first, what is the information relevant to the problems of interest; secondly, how to codify the relevant information into a probability distribution; and third, how to obtain estimates for various quantities of interest? The identification of the relevant information is at present a matter of trial and error; beyond experience and intuition, there exists no systematic method. The assignment of a probability distribution is commonly carried out using Jaynes' method of maximum entropy—MaxEnt. The main goal of this thesis is to use a form of the method of maximum entropy—the ME method, which includes MaxEnt as a special case—to address the third problem.; The basic idea is to select a family of trial probability distributions and use the ME method to select the member of the family which best approximates the “exact” but intractable distribution. Thus we have developed a method for generating approximations which includes, but goes beyond, the well-known variational method of Bogoliubov.; Our first goal is to systematize the method of maximum entropy into what one might call an ME algorithm. The algorithm follows a sequence of steps: identifying relevant information; codifying this information into a probability distribution; selecting an optimal approximate but calculable distribution from within a family of trial distributions; and finally, improving the approximation by also taking into account the full family of trial distributions.; To test the applicability of the method we apply it to a system that has been extensively studied in the past: the theory of simple fluids. We develop approximations based on two families of trial distributions. The first brief treatment leads to a mean-field approximation. The second, more extended treatment, is an approximation in terms of hard spheres. Numerical calculations of the radial distribution function and the thermodynamic properties of Argon are compared to experimental results, to the results of molecular dynamics simulations, and to the results of various perturbation theories.; Our work indicates that the ME-improved variational approach developed here offers predictions that are already competitive with the best variational theories. The potential of the ME approach for further improvements and further applications is, at this early stage, far from being exhausted.
机译：物理学理论总是根据“演绎”的推理方法发展起来的。然而，从热力学到信息论的熵研究显示出一种自然的推理方法，即“推论”，更适合于统计物理学领域。在统计物理学中进行统计预测需要解决三个问题：首先，与感兴趣的问题相关的信息是什么？其次，如何将相关信息整理成概率分布；第三，如何获得各种兴趣量的估计？目前，有关信息的识别是一个反复试验的问题；除了经验和直觉之外，没有系统的方法。通常使用Jaynes最大熵方法MaxEnt来进行概率分布的分配。本文的主要目标是使用最大熵方法的一种形式（ME方法，其中包括MaxEnt作为特例）来解决第三个问题。基本思想是选择一个试验概率分布族，并使用ME方法选择最接近“精确”但难以处理的分布的族成员。因此，我们开发了一种生成近似值的方法，该方法包括但不限于Bogoliubov的众所周知的变分方法。我们的首要目标是将最大熵的方法系统化为所谓的ME算法。该算法遵循一系列步骤：识别相关信息；将这些信息编码为概率分布；从一系列试验分布中选择一个最佳的近似但可计算的分布；最后，通过考虑整个试验分布系列来改善近似值。为了测试该方法的适用性，我们将其应用于过去已经广泛研究的系统：简单流体理论。我们根据两个试验分布族得出近似值。最初的简短处理导致平均场近似。第二种，更广泛的治疗，是硬球的近似值。将氩气的径向分布函数和热力学性质的数值计算与实验结果，分子动力学模拟的结果以及各种微扰理论的结果进行了比较。我们的工作表明，此处开发的经ME改进的变分方法可提供与最佳变分理论相竞争的预测。在早期阶段，ME方法具有进一步改进和进一步应用的潜力，还远远不够。

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  • 作者

    Tseng, Chih-Yuan.;

  • 作者单位

    State University of New York at Albany.;

  • 授予单位 State University of New York at Albany.;
  • 学科 Physics Condensed Matter.; Physics Fluid and Plasma.
  • 学位 Ph.D.
  • 年度 2004
  • 页码 174 p.
  • 总页数 174
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 等离子体物理学;
  • 关键词

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