Helmholtz Reciprocity 1

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:

  • 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

  • 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.


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.


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