Snyder W C, Wan Z, Li X
Appl Opt. 1998 Jun 1;37(16):3464-70. doi: 10.1364/ao.37.003464.
Contrary to common belief, neither reciprocity of the bidirectional reflectance distribution function (BRDF) nor the directional form of Kirchhoff's electromagnetic radiation law can be demonstrated on the basis of energy conservation. The BRDF is generally considered reciprocal as an extension of Helmholtz reciprocity, but Helmholtz reciprocity does not always hold. We describe the flaw in a thermodynamic demonstration of reciprocity that uses an enclosure calculation. Some conclusions can be drawn from the enclosure calculation, but reciprocity requires more restrictive conditions. We conclude that, although they can be violated, reciprocity and the directional form of Kirchhoff's law generally hold because of the quantum-mechanical principle of time-reversal invariance, which applies to most materials.
与普遍看法相反，基于能量守恒既无法证明双向反射分布函数（BRDF）的互易性，也无法证明基尔霍夫电磁辐射定律的方向性形式。BRDF通常被视为亥姆霍兹互易性的扩展而被认为是互易的，但亥姆霍兹互易性并非总是成立。我们描述了一种使用封闭计算的互易性热力学证明中的缺陷。从封闭计算中可以得出一些结论，但互易性需要更严格的条件。我们得出结论，尽管互易性和基尔霍夫定律的方向性形式可能会被违反，但由于适用于大多数材料的时间反演不变性的量子力学原理，它们通常是成立的。
