Quality of Heat Energy vs Radiative Heat Transfer

One may expect that radiative heat transfer as a basic topic of both fundamental and engineering physics, has been well understood since long and that therefore there are precise mathematical models capturing everything of interest from both theoretical and practical point of view. 

But that does not seem to be the case. The basic model (used in climate science) takes the form of the more than 100 years old Schwarzschild's equations describing radiative heat transfer in terms of streams of particles in different directions carrying energy packest named photons between layers of a gas of different temperature, in the setting of the atmosphere with with both up-welling and down-welling infrared radiation. In particular, the equations express transfer of heat energy from colder layers to warmer layers. 

But we know that the 2nd Law of Thermodynamics says that heat energy cannot by itself (without external forcing) move from cold to warm, and so Schwarzschild's equations are unphysical. If they form the mathematical model of radiative heat transfer, it seems that some understanding is missing. I make an effort to fill this apparent gap as Computational Black Body Radiation.

The 2nd Law is believed to be difficult to understand lacking a good answer to the basic question: Why can heat energy by itself move only from warm to cold and not the other way? 

One way to understand this is to think of quality of energy measured by temperature. The higher temperature the better quality. It can be compared with quality of knowledge. The essential aspect is that transfer of heat energy or knowledge has a direction from higher quality to lower quality,  not the other way around. 

The quality of heat energy as measured by temperature comes to expression as a high frequency cut-off of the radiation spectrum scaling with temperature according to Wien's Law as a threshold phenomenon.  A warm body carries higher frequencies than a colder body and it is these frequencies which can cause a transfer of heat energy from the warm to the cold.  A more knowledgeable person can transfer knowledge to a less knowledgeable. There is a direction from high to lower quality, which in more general terms is direction of time.  

In the hierarchy of energy, heat energy has lowest rank because it cannot fully be transformed to higher ranks of kinetic or potential energy, and is further decreasing in rank with decreasing temperature. This underlies a feared successive degradation of quality into a heat death of the Universe at 3 Kelvin. This can be added to the alarm of CO2 global warming.