Radiative Heat Transfer: Resonance

In a sequence of posts I have compared a wave model and a particle model for radiative heat transfer, which can be illustrated in the interaction between the two partners of a marriage:

Wave model:

• Interchange of energy by a phenomenon of resonance.
• The wiser partner transfers energy to the less wise by resonance.
• Happy educated marriage.

Particle model:

• Interchange of energy by invectives.
• The partner with the stronger invectives wins the battle.
• Unhappy primitive marriage.

Let us now add some mathematics to this picture:

Wave model:

The model for radiative exchange of heat energy between two blackbodies 1 and 2 analyzed in Mathematical Physics of Blackbody Radiation, satisfies the following energy balance for each frequency $\nu$:

• $\frac{dOE_1}{dt} + R_1 + IE_1 = F^2 = \frac{dOE_2}{dt} + R_2+ IE_2$

where each blackbody consists of an oscillator with energy $OE$ subject to damping from outgoing radiation of energy $R$ per unit time and damping from internal heating (friction)

of energy $IE$, with the subindex indicating the energy for each body, and $F^2$ representing

the common shared energy.

The damping mechanisms are subject to the following cut-off rule depending on frequency $\nu$ and temperature $T$, assuming a principal form:

• if $\nu > T $ then $R(\nu ) =0$
• if $\nu < T $ then $IE(\nu ) =0$.

The cut-off rule expresses that the outgoing radiation is replaced by internal heating as the frequency passes a threshold proportional to temperature.

We consider a stationary state with $OE_1$ and $OE_2$ constant, in which case the energy balance takes the form

• $R_1 + IE_1 = F^2 = R_2+ IE_2$.

By the cut-off rule it follows that if $T_1>T_2$ then

• R_1 = IE_2

which expresses that the radiation from 1 is stored in 2 as heat, that is radiative transfer from hot to cold in accordance with the 2nd law of thermodynamics.

This model for radiative transfer can be illustrated in the above picture of two harmonic oscillators (blackbodies) connected by a spring (radiation) as a transfer of energy between the oscillators by the connecting spring.

Particle model:

In a particle model of radiative transfer each blackbody is supposed to both emit and absorb photons with the hotter body emitting more than absorbing and thus losing heat and the colder body absorbing more than emitting and thus heating up. This is a primitive model without the physics of resonance.

A model is primitive if the essential physics is missing. A primitive model does not help understanding and has little predictive capability. Primitive is not the same as simple.

A simple model describing essential physics (e.g. Newton's 2nd law) is not primitive and thus may be useful.

CO2 alarmism is based on a particle model for radiative heat transfer supposed supporting

an idea of DLR/backradiation. This is primitive.