The Essence of Planck's Radiation Law and Its Proof

Planck trying to convince a disbelieving Bohr that his proof of Planck's Law is correct. On the blackboard: Maxwell's equations.

The derivation of Planck's Law (of blackbody radiation) given by Planck is based on particle statistics borrowed from thermodynamics with little connection to the real physics of radiation as interaction of matter and electromagnetic wave motion.

Planck was forced into particle statistics because a radiation law based on wave motion seemed to lead a Rayleigh-Jeans law with an "ultraviolet catastrophe" with energy exploding to infinity like $\nu^2$ with frequency $\nu$ without upper bound. By a statistical assumption that highly energetic waves are rare Planck gave physics a way to avoid the catastrophe which earned almost infinite fame and a Nobel Prize.

But Planck left a question without answer: Is it possible to derive a correct radiation law without resort to statistics, using instead classical electrodynamics described by Maxwell's wave equations?

In Mathematical Physics of Blackbody Radiation and Computational Blackbody Radiation I show that this is indeed possible, by replacing statistics by a much more basic physical assumption of finite precision computation. Let me here presents the essence of my argument:

I consider mathematical 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. $U_{tt} - U_{xx}$: material force from vibrating string with U displacement
2. $- \gamma U_{ttt}$ is Abraham-Lorentz (radiation reaction) force
3. $- \delta^2U_{xxt}$ is a viscous force
4. $f$ is exterior forcing,

and $\gamma$ and $\delta^2$ are (small) positive coefficients subject to a frequency dependent switch from outgoing radiation with to absorption/internal heating defined below.

The finite precision computation is represented by the $\delta^2$ viscosity which sets a smallest scale in space of size $\delta$ by damping frequencies higher than $\frac{1}{\delta}$.

The coefficients $\gamma$ and $\delta$ are chosen so that only one is positive for a certain frequency $\nu$ in a spectral decomposition, with a switch from $\gamma >0$ to $\delta >0$ at $\nu = \frac{T}{h}$, where $T$ is temperature and $h$ is a fixed precision parameter representing atomic dimension in the string of atoms modeled by the wave equation.

The switch thus defines a temperature dependent cut-off frequency $\frac{T}{h}$ with the effect that only frequencies below the cut-off are emitted as radiation while frequencies above cut-off are stored as internal heat energy of the string.

The essence of the proof is the following energy balance established by a spectral analysis assuming periodicity in space and time:

• $\int \gamma U_{tt}^2dxdt + \int \delta^2U_{xt}^2dxdt = \epsilon \int f^2 dxdt$

with $\epsilon\approx 1$ is a coefficient of emissivity (= absorptivity), which we write in condensed form as

• $R + A = F$, or in a spectral decomposition $R_\nu + A_\nu = F_\nu$, or in words
• Radiation + Absorption = Forcing,

where $R = \int \gamma U_{tt}^2dxdt$, $A=\int \delta^2U_{xt}^2dxdt$ and $F=\epsilon \int f^2 dxdt$.

As just said, the switch effectively means that

• $R_\nu = F_\nu$ for $\nu <\frac{T}{h}$
• $A_\nu = F_\nu$ for $\nu >\frac{T}{h}$,

which expresses that the forcing $F_\nu$ is remitted as outgoing radiation for $\nu<\frac{T}{h}$, and is absorbed and stored as internal heat energy for $\nu > \frac{T}{h}$.

The coefficient $\epsilon$ represents emissivity = absorptivity, in accordance with Kirchhoff's Law. An essential aspect is that $\epsilon$ is independent of $\gamma$ and $\delta$, which expresses that radiation (to up to the constant $\epsilon$) is independent of the nature of the radiating/absorbing body.

Planck's law in the form $R+A = F$ with only one of $R$ and $A$ non-zero for each frequency

is a variant of Planck's classical law with a sharp switch instead of a continuous transition from radiation to absorption/heating.

Planck's law in the form $R + A = F$ thus contains the classical law for radiation as $R=F$ but also the further information that absorption into internal heat energy $A=R$ only occurs for frequencies above cut-off.

The proof that $R + A = F$ is non-trival and expresses a fundamental aspect of near-resonance

in systems of oscillators with small damping. The proof shows Planck's Law $R_\nu\sim \gamma T\nu^2$ for $\nu <\frac{T}{h}$ and Kirchhoff's Law, without resort to statistics.

The effects of Planck's statistics in modern physics are subject to an investigation in Dr Faustus of Modern Physics: Is it true that Planck's technique for avoiding the ultraviolet catastrophe led to a much bigger catastrophe of abandoning rationality in physics?