用Dimensional Analysis原理推导黑体辐射定律
Heuristic Derivation of Blackbody Radiation Laws using Principles of Dimensional Analysis
摘要：
本文利用量纲分析的原理导出了维恩位移定律和黑体辐射定律的推广形式，以及（a）瑞利和金斯定律，（b）瑞利定律，（c）维恩和帕邢定律，（d）泰森定律和（e）普朗克定律。这种缩放以严格的数学方式表示，采用维度π不变式分析，有时称为白金汉π定理。它示出，在瑞利和牛仔裤的经典辐射定律的情况下，只有一个π数发生，必须被认为是一个无量纲的普遍常数。该pi数可以在理论上或/和经验上确定。它还表明，维π-不变量分析产生一个广义形式的维恩位移定律。在这种情况下，通常会出现两个π数。因此，建立了一个通用的功能，是必不可少的，以避免所谓的瑞利-金斯灾难的紫外线。不幸的是，这样一个通用的功能不能推断出从尺寸参数。它也必须从理论上或/和经验上推导出来。它表明，这样的相似性函数可以推导出启发式原则的基础上，当标准，如最大条件关于广义形式的维恩的位移法，斯特凡和玻尔兹曼的幂律的要求，和埃菲斯特的参数关于红色和紫色的要求。
英文摘要：
A generalized form of Wien's displacement law and the blackbody radiation laws of (a) Rayleigh and Jeans, (b) Rayleigh, (c) Wien and Paschen, (d) Thiesen and (e) Planck are derived using principles of dimensional analysis. This kind of scaling is expressed in a strictly mathematical manner employing dimensional pi-invariants analysis sometimes called Buckingham's pi-theorem. It is shown that in the case of the classical radiation law of Rayleigh and Jeans only one pi number occurs that has to be considered as a non-dimensional universal constant. This pi number may be determined theoretically or/and empirically. It is also shown that dimensional pi-invariants analysis yields a generalized form of Wien's displacement law. In this instance two pi numbers generally occur. Consequently, a universal function is established that is indispensable to avoid the so-called Rayleigh-Jeans catastrophe in the ultraviolet. Unfortunately, such a universal function cannot be inferred from dimensional arguments. It has to be derived theoretically or/and empirically, too. It is shown that such a similarity function can be deduced on the basis of heuristic principles, when criteria like the maximum condition regarding the generalized form of Wien's displacement law, the requirement of the power law of Stefan and Boltzmann, and Ehrenfest's arguments regarding the red and the violet requirements are adopted.
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