用于宇宙学数据分析的广义可组合熵
Extensive Composable Entropy for the Analysis of Cosmological Data
AI总结：
针对Bekenstein-Hawking熵违背三维系统热力学广延性的问题，基于群论提出兼具广延性与可组合性的广义熵泛函，验证了特定参数下其与多类宇宙学数据的一致性。
摘要：
近几十年来，全球范围内大量研究工作基于熵方法聚焦于黑洞与宇宙（如暗能量现象）研究。基于玻尔兹曼-吉布斯统计的Bekenstein-Hawking熵$S_{BH}\propto\boldsymbol{A}/l_P^2$（$A$代表面积；$l_P$代表普朗克长度）始终发挥着关键理论作用，但它存在一个严重缺陷，即违背三维空间系统的热力学广延性。不过其引人关注的面积依赖特性表明，有必要考虑$W(N)\boldsymbol{\sim}\mu^{N^\gamma}\\;\\;(\mu>1;\gamma>0)$的形式，其中$W$和$N$分别为微观状态总数与组分数；$\gamma=1$对应标准的玻尔兹曼-吉布斯（BG）统计力学。针对这一$W(N)$渐近行为，我们基于群论提出了熵泛函$S_{\alpha,\gamma}=k \Bigl[ \frac{\ln \Sigma_{i=1}^W p_i^\alpha}{1-\alpha} \Bigr]^{\frac{1}\gamma} \\\\;(\alpha\in \mathbb{R};\\,S_{1,1}=S_{BG}\equiv-k\sum_{i=1}^W p_i \ln p_i)$。该泛函对任意$(\alpha,\gamma)$均同时满足广延性（热力学要求）与可组合性（逻辑自洽性要求）。我们进一步证明，当$(\alpha,\gamma)=(1,2/3)$时，该熵与中微子测量、大爆炸核合成、冷暗物质粒子遗迹丰度的宇宙学数据，以及动力学和几何宇宙学数据集均吻合良好。
英文摘要：
Along recent decades, an intensive worldwide research activity is focusing both black holes and cosmos (e.g. the dark-energy phenomenon) on the basis of entropic approaches. The Boltzmann-Gibbs-based Bekenstein-Hawking entropy $S_{BH}\propto A/l_P^2$ ($A \equiv$ area; $l_P \equiv$ Planck length) systematically plays a crucial theoretical role although it has a serious drawback, namely that it violates the thermodynamic extensivity of spatially-three-dimensional systems. Still, its intriguing area dependence points out the relevance of considering the form $W(N)\sim μ^{N^γ}\;\;(μ>1;γ>0)$, $W$ and $N$ respectively being the total number of microscopic possibilities and the number of components; $γ=1$ corresponds to standard Boltzmann-Gibbs (BG) statistical mechanics. For this $W(N)$ asymptotic behavior, we introduce here, on a group-theory basis, the entropic functional $S_{α,γ}=k \Bigl[ \frac{\ln Σ_{i=1}^W p_i^α}{1-α} \Bigr]^{\frac{1}γ} \;(α\in \mathbb{R};\,S_{1,1}=S_{BG}\equiv-k\sum_{i=1}^W p_i \ln p_i)$. This functional simultaneously is {\it extensive} (as required by thermodynamics) and {\it composable} (as required for logic consistency), $\forall (α,γ)$. We further show that $(α,γ)=(1,2/3)$ satisfactorily agrees with cosmological data measuring neutrinos, Big Bang nucleosynthesis and the relic abundance of cold dark matter particles, as well as dynamical and geometrical cosmological data sets.
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