双参数复杂性与广义普朗克辐射定律
Biparametric complexities and the generalized Planck radiation law
摘要：
复杂性理论体现了现代科学中一些最困难、最基本、最具挑战性的开放问题。复杂性这个术语非常难以捉摸，所以这个理论的主要目标是为它找到有意义的量词。事实上，我们需要各种各样的度量来考虑这个术语的多个方面。这里定义并讨论了连续概率分布复杂性的一些双参数Crámer-Rao和Heisenberg-Rényi测度。然后，他们被应用到黑体辐射在温度T在d维宇宙。发现这些无量纲量既不依赖于T，也不依赖于任何物理常数。因此，它们具有普适性，即它们只依赖于空间维度。为了确定这些复杂性量词，我们计算了它们的分散（典型偏差）和熵（雷尼熵和广义费舍尔信息）成分。他们被发现有一个温度依赖性的行为类似于著名的维恩位移法的主频$ν_{max}$，在频谱达到最大值。此外，它们使我们能够深入了解d维黑体光谱的新方面，以及与空间维度相关的量子效应的量化。
英文摘要：
Complexity theory embodies some of the hardest, most fundamental and most challenging open problems in modern science. The very term complexity is very elusive, so that the main goal of this theory is to find meaningful quantifiers for it. In fact we need various measures to take into account the multiple facets of this term. Here some biparametric Crámer-Rao and Heisenberg-Rényi measures of complexity of continuous probability distributions are defined and discussed. Then, they are applied to the blackbody radiation at temperature T in a d-dimensional universe. It is found that these dimensionless quantities do not depend on T nor on any physical constants. So, they have an universal character in the sense that they only depend on the spatial dimensionality. To determine these complexity quantifiers we have calculated their dispersion (typical deviations) and entropy (Rényi entropies and the generalized Fisher information) constituents. They are found to have a temperature-dependent behavior similar to the celebrated Wien's displacement law of the dominant frequency $ν_{max}$ at which the spectrum reaches its maximum. Moreover, they allow us to gain insights into new aspects of the d-dimensional blackbody spectrum and about the quantification of quantum effects associated with space dimensionality.
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