Model of One Blackbody
In our series on blackbody radiation we have mostly focussed on radiative equilibrium, and so let us now consider the more general question of the dynamics of radiative interaction of two blackbodies of different temperature.
We recall our model for one blackbody subject to radiative forcing, explored in Mathematical Physics of Blackbody Radiation and Computational Blackbody Radiation, as a wave equation expressing force balance:
where the subindices indicate differentiation with respect to space and time , and
- is out-of-equilibrium force of a vibrating string with displacement ,
- is the Abraham-Lorentz (radiation reaction) force with a small positive parameter,
- is a friction force replacing the radiation reaction force for frequencies larger than a cut-off frequency and then contributing to the internal energy,
- is a smallest coordination length with a measure of finite precision,
- is the common energy/temperature of each frequency of the vibrating string,
- is exterior forcing.
The model is specified by the parameters and . It is shown in Universality of Blackbody Radiation that all blackbodies can be assumed to have the same value of the radiation coefficient and the cut-off (precision ), given as the values of a chosen reference blackbody with the property that is maximal and minimal.
Stationary periodic solutions satisfy the energy balance
which expresses that all incident radiation is absorbed and is either re-emitted as radiation or stored as internal energy from heating with a switch from to at the cut-off frequency. We here assume that all frequencies have the same energy , where is the amplitude of frequency , and we refer to the common value as the temperature.
With dynamics the wave equation (1) expressing force balance is complemented by an equation for the total energy :
expressing that the change is balanced by the outgoing radiation and absorbed energy from the forcing , where with the string energy and the internal energy as accumulated dissipated energy . Equivalently, the change of the internal energy is given as
The temperature connects to by , assuming all frequencies of have the same energy , because
assuming being dominated by .
Our model thus consists of (1) + (2) combined with a mechanism for equidistribution of energy over all frequencies.
Radiative Interaction of Two Blackbodies
Let us now consider two blackbodies in radiative contact, one body with amplitude sharing a common forcing with another blackbody with amplitude , modeled by the wave equation:
- and .
This system describes the dynamic interaction of and , with given initial values of , and for and the same for .
In dynamic interaction different frequencies will have different times scales and thus to maintain that all frequencies have the same temperature some mechanism to this effect will have to be adjoined to the model. We may think of this effect as a form of diffusion acting on frequencies.
We compare with the case of acoustic damping with an acoustic damping term of the form in which case all frequencies will have the same damping and thus well tempered distributions will be preserved under dynamic interaction. Recall that a piano as a blackbody with acoustic damping is isotempered in the sense that the sustain of different tones is the same.
Let us now see what the model tells in different basic cases:
1. Both below cut-off.
Let us now consider the basic case of interaction with all frequencies below cut-off for both and . The difference then satisfies the damped wave equation
which upon multiplication by and integration in space and time gives for (modulo two terms from integration by parts in time with small effect if the time scale is not short):
It follows that decays in time which effectively means equilibration in energy with a transfer of energy from the warmer to the colder body.
The effect of the dissipative radiation term () is that the difference in energy (and thus temperature) between the two bodies decreases with time: Energy is transferred from the warmer to the colder body. We have thus proved a 2nd Law as an effect of dissipative radiation.
2. One below one above cut-off
For frequencies below cut-off for and above cut-off for , assuming is the warmer, the model shows a transfer from into with a heating effect on as a result of the energy balance
3. Both above cut-off
This is analogous to case 1.