Stefan-Boltzmann定律之谜：经典还是量子物理
The Puzzling of Stefan-Boltzmann Law: Classical or Quantum Physics
摘要：
Stefan-Boltzmann定律由Stefan在1874年通过拟合现有实验经验地推导出，并由Boltzmann在1884年基于涉及热力学原理和麦克斯韦方程组的经典模型在理论上验证。乍看之下，玻尔兹曼假设的电磁气体是$N$经典简正模的系综，看起来像是大质量理想气体经典模型的延伸。因此，对于这种EM气体，内部总能量$U$被认为是体积$V$和温度$T$的函数，作为$U=U(V,T)$，状态方程由$U=3PV$给出，其中$P$是辐射压力。此外，玻尔兹曼隐含地假设，对于给定的$V$和$T$值，$U$和$N$将取有限值。然而，从一方面来看，这些假设并不被麦克斯韦方程组所证明，因为在真空中（即远离电磁源），根据经典统计，$U$和$N$的值是发散的。另一方面，玻尔兹曼导出的斯蒂芬定律与普朗克1901年宣布的量子统计导出的斯蒂芬定律在宏观上是相容的。因此，这封信提出了一个解决这个令人困惑的经典/量子兼容性的方法，注意到玻尔兹曼所做的隐式假设完全被量子统计所证明。此外，我们揭示了新的解释，最近的经典模拟的黑体由王，Casati，和Benenti在2022年谁发现了类似的令人困惑的兼容性，以诱导对经典物理和黑体辐射的猜测，声称需要一个批判性的重新考虑的作用，经典物理学的理解量子力学。
英文摘要：
Stefan-Boltzmann law was empirically deduced by Stefan in 1874 by fitting existing experiments and theoretically validated by Boltzmann in 1884 on the basis of a classical model involving thermodynamics principles and the Maxwell equations. At first sight the electromagnetic (EM) gas assumed by Boltzmann and identifiable as an ensemble of $N$ classical normal-modes, looks like an extension of the classical model of the massive ideal-gas. Accordingly, for this EM gas the internal total energy, $U$, was taken to be function of volume $V$ and temperature $T$ as $U=U(V,T)$, and the equation of state was given by $U=3PV$, with $P$ the radiation pressure. In addition, Boltzmann implicitly assumed that, for given values of $V$ and $T$, $U$ and $N$ would take finite values. However, from one hand these assumptions are not justified by Maxwell equations since, in vacuum (i.e. far from the EM sources), according to classical statistics, the values of $U$and $N$ diverge. From another hand, Boltzmann derivation of Stefan law is found to be macroscopically compatible with its derivation from quantum statistics announced by Planck in 1901. Accordingly, this letter presents a solution of this puzzling classical/quantum compatibility by noticing that the implicit assumption made by Boltzmann is fully justified by quantum statistics. Furthermore, we shed new light on the interpretation of recent classical simulations of a black-body carried out by Wang, Casati, and Benenti in 2022 who found an analogous puzzling compatibility to induce speculations on classical physics and black-body radiation that are claimed to require a critical reconsideration of the role of classical physics for the understanding of quantum mechanics.
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