Maximum Entropy Methods as the Bridge Between Microscopic and Macroscopic Theory
Maximum Entropy Methods as the Bridge Between Microscopic and Macroscopic Theory
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DOI：
10.1007/s10955-016-1587-8
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发表时间：
2015-02
影响因子：
1.6
通讯作者：
Jamie M. Taylor
中科院分区：
文献类型：
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作者：
Jamie M. Taylor
本文以Ball和Majumdar先前的工作为基础，对称为奇异势的宏观变量函数进行了研究。奇异势是状态空间上概率分布的可容许统计平均的函数，定义为它对应于给定已知观测到的统计平均的最大可能熵，尽管非经典类熵目标函数也将被考虑。首先必须建立容许矩集，在本工作中提出的条件下，容许矩集是开放的、有界的和凸的，允许用支持的超平面来描述，这提供了对相关概率分布奇点发展的估计。在适当的条件下，证明了奇异势是严格凸的，与微观熵一样可微，并在宏观变量趋于容许矩集边界时均匀爆炸。然后讨论奇异势的应用，并特别考虑平均场理论中典型的某些自由能泛函，证明某些微观和宏观自由能泛函之间的等价性。这允许得到关于Onsager自由能的局部极小值的陈述，这是不能由双边变化给出的，并且克服了确保局部极小值在取变化之前远离零的需要。分析还允许定义一个双阶参数，其中Onsager自由能允许显式表示。此外，讨论了用处处定义的函数，特别是多项式函数逼近奇异势的困难，并举例说明了泰勒近似在保持奇异势的相关形状性质方面的失败。
This paper is concerned with an investigation into a function of macroscopic variables known as the singular potential, building on previous work by Ball and Majumdar. The singular potential is a function of the admissible statistical averages of probability distributions on a state space, defined so that it corresponds to the maximum possible entropy given known observed statistical averages, although non-classical entropy-like objective functions will also be considered. First the set of admissible moments must be established, and under the conditions presented in this work the set is open, bounded and convex allowing a description in terms of supporting hyperplanes, which provides estimates on the development of singularities for related probability distributions. Under appropriate conditions it is shown that the singular potential is strictly convex, as differentiable as the microscopic entropy, and blows up uniformly as the macroscopic variable tends to the boundary of the set of admissible moments. Applications of the singular potential are then discussed, and particular consideration will be given to certain free-energy functionals typical in mean-field theory, demonstrating an equivalence between certain microscopic and macroscopic free-energy functionals. This allows statements about-local minimisers of Onsager’s free energy to be obtained which cannot be given by two-sided variations, and overcomes the need to ensure local minimisers are bounded away from zero andbefore takingvariations. The analysis also permits the definition of a dual order parameter for which Onsager’s free energy allows an explicit representation. Also, the difficulties in approximating the singular potential by everywhere defined functions, in particular by polynomial functions, are addressed, with examples demonstrating the failure of the Taylor approximation to preserve relevant shape properties of the singular potential.