Difference Between Emission and Absorption of Radiation

In a sequence of posts on radiative heat transfer and DLR/backradiation I have studied a wave model of the form:

• $U_{tt} - U_{xx} - \gamma U_{ttt} - \delta^2U_{xxt} = f $

where the subindices indicate differentiation with respect to space $x$ and time $t$, and

1. $U_{tt} - U_{xx}$ represents a vibrating string with U displacement
2. $- \gamma U_{ttt}$ is a dissipative term modeling outgoing radiation = emission
3. $- \delta^2U_{xxt}$ is a dissipative modeling internal heating = absorption
4. $f$ is incoming forcing/microwaves,

where $\gamma$ and $\delta^2$ are positive constants connected to dissipative losses as outgoing radiation = emission and internal heating = absorption.

We see emission represented by $-\gamma U_{ttt}$ and absorption by $-\delta^2U_{xxt}$. We now ask:

1. How is the distinction between emission and absorption expressed in this model?
2. Is Helmholtz Reciprocity valid (emission and absorption are reverse processes)?
3. Is Kirchhoff's Radiation Law (emissivity = absorptivity) valid?

Before seeking answers let us recall the basic energy balance between incoming forcing $f$ measured as

• $F = \int f^2(x,t)\, dxdt$

assuming periodicity in space and time and integrating over periods, and (rate of) outgoing radiation = emission $R$ measured by

• $R = \int \gamma U_{tt}^2\, dxdt$,

the oscillator energy $OE$ measured by

• $OE =\frac{1}{2}\int (U_t^2 + U_x^2)\, dxdt$

and (rate of) internal energy = absorption measured by

• $IE = \int \delta^2U_{xt}^2\, dxdt$

with the energy balance in stationary state expressed as

• $F = R + IE$
• incoming energy = emission + absorption.

The model has a frequency switch switching from emission to absorption as the frequency increases beyond a certain threshold proportional to temperature in accordance with Wien's displacement law.

We now return to questions 1 - 3.

Both terms generate dissipative effects when multiplied with $U_t$ as $R = \int \gamma U_{tt}^2\, dxdt$ and $IE = \int \delta^2U_{xt}^2\, dxdt$, but the terms involve different derivatives with $U_{tt}$ acting only in time and $U_{xt}$ acting also in space.

The absorption $U_{xt}^2$ represents a smoothing effect in space, which is irreversible and thus cannot be reversed into emission as reversed absorption.

The emission $U_{tt}^2$ represents a smoothing effect in time, which is irreversible and thus cannot be reversed into absorption as reversed emission.

In other words, in the model both absorption and emission are time irreversible and thus cannot be reversed into each other.

We conclude that the model does not satisfy Helmholtz reciprocity.

Nevertheless, the model satisfies Kirchhoff's law as shown in a previous post.

Conclusion:

The space derivative in $U_{xt}$ models absorption as process of smoothing in space with irreversible transformation of high frequencies in space into low frequencies with a corresponding increase of internal energy as heat energy.

Absorbed high frequencies can with increasing temperature be rebuilt through (resonance in) the wave equation into high frequency emission.

Absorption and emission are not reverse processes, but my be transformed into each other

through (resonance in) the wave equation and the switch.

We may compare absorption with a catabolic process of destroying (space-time) structure and emission with an anabolic process of building structure, with the wave equation as a transformer.

For more details see Mathematical Physics of Blackbody Radiation and The Clock and the Arrow.

Helmholtz Reciprocity and DLR/Backradiation

Helmholtz Reciprocity Principle (HRP) states that absorption and emission of light can be viewed as reversed processes arising by reversal of time. Absorption of a light ray by a (black) body is simply emission of a light ray running backwards in time, or the other way around.

Downwelling Longwave Radiation (DLR) and backradiation seek justification by HRP. Kirchhoff's Radiation Law stating that emissivity and absorptivity of a radiating body are equal,

was justified by Planck with reference to HRP.

But is HRP a valid physical principle? What is the relation between HRP and the 2nd Law?

HRP describes reversible physics while the 2nd Law described irreversible physics, and so HRP and the 2nd Law describe different physics.

Is the real physics of absorption and emission the time reversal of each other? Probably not.

The process of absorption is like reading and the process of emission like writing. To state

that reading and writing are the reverse of each other would miss the difference between the active constructive aspect of reading and the passive consumption aspect of reading.

The analysis in Mathematical Physics of Blackbody Radiation shows that absorption and emission are different processes satisfying a 2nd law which does not allow time reversal, and thus indicates that HRP is not valid.

Without HRP the support of DLR/backradiation evaporates.

Note that HRP conforms to a corpuscular theory of light as photon particles for which time reversal is no problem. But such a theory is capable of describing only simple ray tracing physics and not real physics.

Who Proved of Kirchhoff's Law of Radiation?

Kirchhoff's Radiation Law stating that the emissivity of a radiating body is equal to its absorptivity presented in 1859, initiated an intense study of blackbody radiation by Rayleigh, Jeans, Wien and others leading into Planck's proof of his radiation law opening to the quantum mechanics of modern physics.

Kirchhoff's Law can be seen as a triviality stating that emission equals absorption as an expression of energy balance. But Kirchhoff's Law concerns emissivity and absorptivity as emission and absorption per unit time, and in this setting it is not at all trivial. The question is why a body capable of absorbing radiation and emitting radiation, must absorb and emit at the same rate? Is it because emission and absorption are simply the reverse of each other with emission simply absorption backwards in time?

No, it is not so trivial, because emission and absorption are different physical processes both with an arrow of time which cannot be reversed. Emission and absorption are not the reverse of each other.

In a previous post I sketched a proof of Kirchhoff's Law based on a wave model with radiative damping analyzed in more detail in Mathematical Physics of Blackbody Radiation.

Let us trace the history of the proof of Kirchhoff's Law with Experimenting theory: The proofs of Kirchhoff's Radiation Law before and after Planck by A. Schirrmacher presenting the following story:

1. Kirchhoff (1859): Thought experiments with mirrors, basic thermodynamics.
2. Planck (1906); Heat rays, basic thermodynamics.
3. Hilbert (1912-14): Integral equation, axiomatic method.

The debate about the proof was intense and no winner was elected. Further studies were made by Dirac and Heitler based on quantum mechanics.

In the modern textbook Radiative Heat Transfer by M. Modest, Kirchhoff's law is presented as a triviality:

• It is easy to show that a black surface also emits a maximum amount of radiative energy, i.e., more than any other body at the same temperature. To show this, we use one of the many variations of Kirchhoff's law: Consider two identical black-walled enclosures, thermally insulated on the outside, with each containing a small object—one black and the other one not. After a long time, in accordance with the Second Law of Thermodynamics, both entire enclosures and the objects within them will be at a single uniform temperature.
• This characteristic implies that every part of the surface (of enclosure as well as objects) emits precisely as much energy as it absorbs. Both objects in the different enclosures receive exactly the same amount of radiative energy. But since the black object absorbs more energy (i.e., the maximum possible), it must also emit more energy than the nonblack object (i.e., also the maximum possible).
• By the same reasoning it is easy to show that a black surface is a perfect absorber and emitter at every wavelength.

We see here an example of a common feature of modern science: A question which once caused a heated debate between the giants of science of the time without ever being settled including

• interpretation of quantum mechanics
• d'Alembert's paradox
• Loschmidt's paradox
• 2nd Law of Thermodynamics

eventually is being put aside as being trivial or a no-issue of little interest.

But Kirchhoff's law is not a triviality and it is a fundamental part of the theory of radiation, and therefore a proof is of considerable interest.

PS To the students at MIT Kirchhoff's Law is presented as trivial energy balance.

Radiative Heat Transfer: Kirchhoff's Law in New Light

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:

(1) $U_{tt} - U_{xx} - \gamma U_{ttt} - \delta^2U_{xxt} = f $

where the subindices indicate differentiation with respect to space $x$ and time $t$, and

1. $U_{tt} - U_{xx}$ models a vibrating material string with $U$ displacement
2. $- \gamma U_{ttt}$ is a dissipative term modeling outgoing radiation
3. $- \delta^2U_{xxt}$ is a dissipative term modeling internal heating by friction
4. $f$ is the amplitude of the incoming forcing,
5. $T$ is temperature with $T^2=\int\frac{1}{2}(\vert U_t^2+U_x^2)dxdt$,
6. (1) expresses a balance of forces,

where $\gamma$ and $\delta^2$ are certain small damping coefficients defined by spectral decomposition as follows in a model case:

• $\gamma = 0$ if the frequency $\nu >\frac{1}{\delta}$
• $\delta = 0 $ if the frequency $\nu < \frac{1}{\delta}$,

where $\delta = \frac{h}{T}$ represents a "smallest coordination length" depending on temperature $T$ and $h$ is a fixed smallest mesh size (representing some atomic dimension).

This represents a switch from outgoing radiation to internal heating as the frequency $\nu$ passes the threshold $\frac{T}{h}$, with the threshold increasing linearly with $T$.

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 $f$ measured as

• $F = \int f^2(x,t)\, dxdt$

assuming periodicity in space and time and integrating over periods, and (rate of) outgoing radiation $R$ measured by

• $R = \int \gamma U_{tt}^2\, dxdt$,

and (rate of) internal energy measured by

• $IE = \int \delta^2U_{xt}^2\, dxdt$,

together with the oscillator energy

• $OE =T^2 = \frac{1}{2}\int (U_t^2 + U_x^2)\, dxdt$

with the energy balance in stationary state with $OE$ constant taking the form

• $F = \kappa (R + IE)$

with $\kappa\lessapprox 1$ is a constant independent of $T$, $\gamma$, $\delta$ and $\nu$. In other words,

• incoming energy = $\kappa\times$ outgoing radiation energy for $\nu <\frac{1}{\delta}$
• incoming energy = $\kappa\times$ stored internal energy for $\nu > \frac{1}{\delta}$,

which can be viewed as an expression of Kirchhoffs' law that emissivity equals absorptivity.

The equality results from the independence of the coefficient $\kappa$ of the damping coefficients $\gamma$ and $\delta^2$, 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 ($\gamma$ and $\delta^2$).

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.