The theory of the Energy Distribution Law of the Normal Spectrum

14 December 1900

This equation has a simple structure for the dependence of the entropy [radiance] of an irradiated monochromatic vibrating resonator on its vibrational energy.

Entropy [radiance] means disorder.

In a completely stationary radiation field, the vibrations of the resonator change their amplitude and phase irregularly.

• This entropy is in that irregularity.
• This is as long as the time intervals are long compared to the period of one vibration, but short compared to the duration of a measurement.

The constant energy of the stationary vibrating resonator can thus only be considered to be a time average.

• It is an instantaneous average of the energies of a many identical resonators which are in the same stationary radiation field, but far enough from one another not to influence each other directly.

Thus, the entropy [radiance] of a resonator is determined by how the energy is distributed at one time over many resonators.

We should evaluate this quantity of entropy by introducing probability into the electromagnetic theory of radiation.

• The importance of this for the second law of thermodynamics* was originally discovered by Mr. L. Boltzmann.

I derived deductively an expression for the entropy [radiance] of a monochromatically vibrating resonator.

• This gave a normal spectrum* (energy distribution in a stationary radiation state).

This equation:

• does not need you to know a previous spectral formula or any theory.
• can let you know the distribution of a given amount of energy over the different colours of the normal spectrum using one constant of nature only
• the temperature of this energy radiation using a second constant of nature.

Let us consider a large number of linear, monochromatically vibrating resonators—N of frequency v (per second),28 N’ of frequency v, N" of frequency v"’,…, with all N large numbers.

These resonators are separated and are enclosed in a diathermic27 medium with light velocity c and bounded by reflecting walls.

The system has total energy £",(erg) which is present partly in the medium as travelling radiation and partly in the resonators as vibrational energy.

In a stationary state, how is this energy distributed:

• over the vibrations of the resonators and
• over the colours of the radiation in the medium?

What will be the temperature of the total system?

The Value of h

We first consider the vibrations of the resonators then assign to them arbitrary energies.

Energy E goes to the N resonators v, E’ to the N’ resonators v , …

The sum E+E’ + E" + … = E0 must, of course, be less than Et.

The remainder Et — E0 pertains then to the radiation present in the medium.

We then give the distribution of the energy over the separate resonators of each group, first, the distribution of the energy E over the N resonators of frequency v.

E is a continuously divisible quantity.

• So this distribution is possible in infinitely many ways.

E is composed of a well-defined number of equal parts.

We use the constant of nature h = 6-55 x 10~27 erg sec.30

This constant multiplied by the common frequency v of the resonators gives us the energy element31 e in erg.

Dividing E by e we get the number P of energy elements which must be divided over the TV resonators.

If the ratio thus calculated is not an integer, we take for P an integer in the neighbourhood.32

The distribution of P energy elements over N resonators can only take place in a finite, well-defined number of ways.

Each of these ways of distribution we call a “complexion”,33 from Boltzmann for a similar concept.

If we denote the resonators by the numbers 1, 2, 3, …, N, and write these in a row, and if we under each resonator put the number of its energy elements, we get for each complexion a symbol of the following form

1 2 3 4 5 6 7 8 9 10
7 38 11 0 9 2 20 4 4 5

We have taken here 7V= 10, P = 100.

The number of all possible complexions is equal to the number of all possible sets of numbers which we can obtain in this way for the lower sequence for given N and P.

Two complexions must be considered to be different if the corresponding sequences contain the same numbers, but in different order.