Annus MirabilisEinstein in 1905
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Read · Light quanta: from entropy to an energy scale · Results

§5 · independent configurations

Einstein argues that light, when it is emitted or absorbed, behaves as if its energy came in separate quanta, and calls the idea a heuristic point of view. Follow all nine sections, from the trouble with classical radiation to the photoelectric effect.

The paper's results

Pages 141, 142

Boltzmann's principle and the entropy of independent points

Entropy and probability are related by S−S0=RNlg⁡WS - S_0 = \frac{R}{N}\lg W, and for nn points moving independently in a volume v0v_0 the probability that at a chosen instant all of them are in the part vv is (v/v0)n(v/v_0)^n, so their entropy depends on the volume as S−S0=R(nN)lg⁡(vv0)S - S_0 = R\left(\frac{n}{N}\right)\lg\left(\frac{v}{v_0}\right).

As printed

Bedeutet S0S_0 die Entropie bei einem gewissen Anfangszustande eines betrachteten Systems und WW die relative Wahrscheinlichkeit eines Zustandes von der Entropie SS, so erhalten wir also allgemein: S−S0=RNlg⁡W.S - S_0 = \frac{R}{N} \lg W.EntropyEntropyGas constantAvogadro's numberProbability of the configurationExplain this equation

Für diese Wahrscheinlichkeit, welche eine „statistische Wahrscheinlichkeit“ ist, erhält man offenbar den Wert: W=(vv0)n;W = \left( \frac{v}{v_0} \right)^n ;Probability of the configurationVolumeVolumeNumber of independent pointsExplain this equation man erhält hieraus durch Anwendung des Boltzmannschen Prinzipes: S−S0=R(nN)lg⁡(vv0).S - S_0 = R \left( \frac{n}{N} \right) \lg \left( \frac{v}{v_0} \right).EntropyEntropyGas constantNumber of independent pointsAvogadro's numberVolumeVolumeExplain this equation

Text on page 141 of the German source, § 5, paragraph 6 · Text on page 142 of the German source, § 5, paragraph 8

In modern notation

W=fnW = f^{n}
ΔSgas=kB ln⁡(W)=n kB ln⁡(f)\Delta S_{\mathrm{gas}} = k_B\,\ln\left(W\right) = n\,k_B\,\ln\left(f\right)
ΔSgas=n kB ln⁡(f)\Delta S_{\mathrm{gas}} = n\,k_B\,\ln\left(f\right)
  • Premise: The value R/NR/N of the constant CC comes from the kinetic theory of gases, as the paper says.
  • Premise: The points move independently of one another; the probability is a product of one factor per point only because of that.

Decoder

WW
The relative probability of a state of entropy SS, compared with the initial state of entropy S0S_0.
nn
The number of points moving independently of one another in the volume v0v_0.
"statistische Wahrscheinlichkeit"
Statistical probability: the probability of finding a configuration at an arbitrarily chosen instant.
R/NR/N
The universal constant CC in S=Clg⁡W+konst.S = C\lg W + \text{konst.}, whose value the paper takes from the kinetic theory of gases.

Try it

LQ-05: With two independent points and half the volume, how often are both found in the left half? Choose the preset “First encounter (n = 2, f = 1/2)” there.

LQ-05: Lock the points' positions together: does the probability still come out as a power of the volume ratio? Choose the preset “The locked positions (n = 10, locked counterexample)” there.

Where this stops: Einstein says he will show in a separate paper that statistical probability is enough; this paper does not give that proof, and this passage does not supply it. A moment chosen at random presupposes a system in a steady condition, watched over a long time; the passage does not treat systems that are not. Read the argument

Where this stops: The locked-position example is an authored mathematical counterexample, not an established alternative theory of radiation. Spontaneous concentration probability and constrained-state entropy are related but are not the same observable. Read the argument

Argument
Boltzmann's principle, applied to independent points
Model
Exact for points that move independently
Evidence
A derivation, not evidence
History
Boltzmann's principle is earlier work; the paper states it in this form and applies it

§5 · independent configurations

Which probability the entropy uses

Einstein uses the probability of a state in one definite sense: how likely it is to find the system in that state at a moment chosen at random. Read that way, Boltzmann's principle ties entropy to probability without first deciding by hypothesis which cases are equally likely.

Read the argument

Independence supplies the exponent

For n independent uniformly distributed points, the chance that all lie in a fraction f of a volume is fⁿ. Perfectly locked positions instead give f; the exponent is a statement about independence.

Read the argument