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 [radiance].

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

• This entropy [radiance] is in that irregularity.

This is as long as the timespans are:

• long compared to the timespan 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 [radiance] 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.