d维黑体辐射的熵和复杂性
Entropy and complexity properties of the d-dimensional blackbody radiation
摘要：
空间维度是分析自然系统和现象的结构和动力学的关键变量。黑体辐射的维数效应是近年来研究的热点。这些研究仍然有些零碎，对各个科学和技术领域提出了难以克服的定性和定量问题。在这项工作中，我们进行了一个信息理论分析的光谱能量密度的d维黑体在温度T的各种熵类量（不平衡，香农熵，费舍尔信息），以及由三个（无量纲）的复杂性措施（Crámer-Rao，费舍尔-香农和LMC）。所有这些频率函数量的计算和讨论的温度和维数。结果表明，这三种复杂性测度都具有普适性质，即它们既不依赖于温度，也不依赖于普朗克常数和玻耳兹曼常量，而只依赖于空间维数d.此外，当d增加时，它们减小;特别地，对于三维黑体辐射的复杂性的Crámer-Rao、Fisher-Shannon和LMC测量，分别发现值2.28415、1.90979和1.17685。此外，除了谱密度最大的频率（遵循著名的维恩位移定律），三个进一步的特征频率被定义在前面的熵量方面;它们被证明服从类维恩定律。物理上讨论了黑体光谱的这些独特特征的潜在用途。
英文摘要：
Space dimensionality is a crucial variable in the analysis of the structure and dynamics of natural systems and phenomena. The dimensionality effects of the blackbody radiation has been the subject of considerable research activity in recent years. These studies are still somewhat fragmentary, pos- ing formidable qualitative and quantitative problems for various scientific and technological areas. In this work we carry out an information-theoretical analysis of the spectral energy density of a d-dimensional blackbody at temperature T by means of various entropy-like quantities (disequilibrium, Shannon entropy, Fisher information) as well as by three (dimensionless) complexity measures (Crámer-Rao, Fisher-Shannon and LMC). All these frequency-functional quantities are calculated and discussed in terms of temperature and dimensionality. It is shown that all three measures of complexity have an universal character in the sense that they depend neither on temperature nor on the Planck and Boltzmann constants, but only on the the space dimensionality d. Moreover, they decrease when d is increasing; in particular, the values 2.28415, 1.90979 and 1.17685 are found for the Crámer-Rao, Fisher-Shannon and LMC measures of complexity of the 3-dimensional blackbody radiation, respectively. In addition, beyond the frequency at which the spectral density is maximum (which follows the well-known Wien displacement law), three further characteristic frequencies are defined in terms of the previous entropy quantities; they are shown to obey Wien-like laws. The potential usefulness of these distinctive features of the blackbody spectrum is physically discussed.
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