Secret of Conductive and Radiative Heat Transfer

This is a continuation of the previous post on Heat Conduction in Solids as Radiative Heat Transfer with  clarifying analysis from Mathematical Physics of Blackbody Radiation and Computational Blackbody Radiation.

The key aspect of both conductive and radiative heat transfer is interaction in a coupled system of weakly damped oscillators of different frequencies tending to an equilibrium with all oscillators having the same temperature as the system temperature. The damping can be frictional (1st order time derivative) or radiative (3rd order time derivative) 

There are two main questions: (i) Why do different systems take on the same temperature? (ii) Why do oscillators with different frequencies in a system take on the same temperature?  

The answer is hidden in the interaction between incoming radiation, oscillator and outgoing radiation in a weakly radiatively damped oscillator analysed in detail in the above texts. The essence is that under near resonance between incoming frequency and oscillator frequency,  

• incoming radiation is balanced by outgoing radiation plus internal heating. 

This is a non-trivial basic fact reflecting that the forcing and oscillator are out-of-phase with a shift of half a period as a consequence of small radiative damping and near resonance. 

Two coupled oscillators thus interact with outgoing from one oscillator acting as incoming for the other and vice versa and so are led to take on the same temperature, which is then spread over the oscillators of a system and also over systems. 

The essential components in this equilibration process are thus

• weakly damped oscillators generating outgoing radiation and internal heating 
• out-of-phase balance between forcing and damping from near resonance
• high-frequency cut-off increasing with temperature from finite precision computation.  

This analysis connects to Planck's derivation of his law of radiation with statistics replaced by finite precision thus replacing non-physics by physics.