Kirchhoff’s Law of Thermal Radiation: 150 Years starts off by:
- Kirchhoff’s law is one of the simplest and most misunderstood in thermodynamics.
Let us see what we can say about Kirchhoff’s Radiation Law stating that the emissivity and absorptivity of a radiating body are equal, in the setting of the wave model with damping presented in Computational Blackbody Radiation and Mathematical Physics of Blackbody Radiation:
where the subindices indicate differentiation with respect to space and time , and
- models a vibrating material string with displacement
- is a dissipative term modeling outgoing radiation
- is a dissipative term modeling internal heating by friction
- is the amplitude of the incoming forcing,
- is temperature with ,
- the wave equation expresses a balance of forces,
where and are certain small damping coefficients defined by spectral decomposition as follows in a model case:
- if the frequency
- if the frequency ,
where represents a “smallest coordination length” depending on temperature and is a fixed smallest mesh size (representing some atomic dimension).
This represents a switch from outgoing radiation to internal heating as the frequency passes the threshold , with the threshold increasing linearly with .
The idea is that a hotter vibrating string is capable of radiating higher frequencies as coherent outgoing radiation. The switch acts as a band filter with frequencies outside the band being stored as internal heat instead of being radiated: The radiator is then muted and heats up internally instead of delivering outgoing radiating.
A spectral analysis, assuming that all frequencies share a common temperature, shows an energy balance between incoming forcing measured as
assuming periodicity in space and time and integrating over periods, and (rate of) outgoing radiation $R$ measured by
and (rate of) internal energy measured by
together with the oscillator energy
with the energy balance in stationary state with constant taking the form
with is a constant independent of , , and . In other words,
- incoming energy = outgoing radiation energy for
- incoming energy = stored internal energy for ,
which can be viewed as an expression of Kirchhoffs’ law that emissivity equals absorptivity.
The equality results from the independence of the coefficient of the damping coefficients and , and frequency.
Summary: The energy of damping from outgoing radiation or internal heating is the same even if the damping terms represent different physics (emission and absorption) and have different coefficients ( and ).
PS: Note that internal heat energy accumulating under (high-frequency) forcing above cut-off eventually will be transformed into low-frequency outgoing radiation, but this transformation is not part of the above model.