Helmholtz Reciprocity states that certain phenomena of propagation of light and interaction between light and matter, are reversible.

Let us study reversibility in our model of blackbody radiation:

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

- represents a vibrating string with U displacement
- is a dissipative term modeling outgoing radiation = emission
- is a dissipative modeling internal heating = absorption
- is incoming forcing/microwaves,

where and are positive constants connected to dissipative losses as outgoing radiation = emission and internal heating = absorption.

We see emission represented by and absorption by . We now ask:

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

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

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

- ,

the** oscillator energy** measured by

and (rate of)** internal energy = absorption** measured by

- 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 as and , but the terms involve different derivatives with acting only in time and acting also in space.

The absorption represents a **smoothing effect in space**, which is irreversible and thus cannot be reversed into emission as reversed absorption.

The emission 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 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.