高斯分布的不可约分解与黑体辐射谱
Irreducible decomposition of Gaussian distributions and the spectrum of black-body radiation
摘要：
本文证明了一个经典混沌场的模的能量，作为一个经典随机变量，服从连续指数分布，可以唯一地分解为它的分数部分和它的整数部分之和。整数部分是一个离散的随机变量（我们称之为普朗克变量），其分布正好是玻色分布，从而产生了普朗克黑体辐射定律。分数部分是具有连续分布的暗部分（我们称之为暗变量），当然，在实验中没有观察到。证明了玻色分布是无限可分的，并给出了它的不可约分解。普朗克变量可以被分解成一个无限多个独立的二元随机变量，它们代表能量为2^s*h*nu的二元光子（更准确地说是光分子或光多重态），其中s= 0，1，2. .这些二元光子遵循费米统计。因此，黑体辐射可以被看作是统计和热力学独立的费米子气体的混合物组成的二元光子。二元光子为任意（但非相干）普通光子激发的并矢展开提供了一个自然的工具。结果表明，二元光子具有费米子的波粒涨落。这些涨落结合起来，给出了由爱因斯坦涨落公式表示的原始玻色子光子的波粒涨落。
英文摘要：
It is shown that the energy of a mode of a classical chaotic field, following the continuous exponential distribution as a classical random variable, can be uniquely decomposed into a sum of its fractional part and of its integer part. The integer part is a discrete random variable (we call it Planck variable) whose distribution is just the Bose distribution yielding the Planck law of black-body radiation. The fractional part is the dark part (we call is dark variable) with a continuous distribution, which is, of course, not observed in the experiments. It is proved that the Bose distribution is infinitely divisible, and the irreducible decomposition of it is given. The Planck variable can be decomposed into an infinite sum of independent binary random variables representing the binary photons (more accurately photo-molecules or photo-multiplets) of energies 2^s*h*nu with s=0,1,2... . These binary photons follow the Fermi statistics. Consequently, the black-body radiation can be viewed as a mixture of statistically and thermodynamically independent fermion gases consisting of binary photons. The binary photons give a natural tool for the dyadic expansion of arbitrary (but not coherent) ordinary photon excitations. It is shown that the binary photons have wave-particle fluctuations of fermions. These fluctuations combine to give the wave-particle fluctuations of the original bosonic photons expressed by the Einstein fluctuation formula.
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