Light quanta · Sections 3-4
A spectrum has an entropy. Compressing it costs the same way a gas does.
Within the regime where Wien's law holds, how does the entropy of monochromatic radiation depend on the volume it occupies, and what had to be fixed to get a definite answer?
Radiation entropy workbench
The radiation entropy workbench
Static worked example
CurrentThese numbers match the current settings.
Model note
- Primary output radiationEnergy: Host calculation (lq04.acceptedInputs). Owner lq04.acceptedInputs.
- Primary outputs initialTemperature, finalTemperature, initialX, finalX, initialSpectralEntropyDensity, finalSpectralEntropyDensity, radiationEntropy, radiationEntropyNumeric, entropyVolumeCoefficient, unfixedConstantDeltaS, unfixedConstantExtraTerm: Host calculation (radiation.entropy). Owner radiation.entropy.
- Primary outputs initialPointwiseDeviation, finalPointwiseDeviation: Host calculation (radiation.spectra). Owner radiation.spectra.
- Primary output effectiveIndependentCount: Host calculation (lq04.derivedFromEnergyAndPlanckConstant). Owner lq04.derivedFromEnergyAndPlanckConstant.
- Accepted input revision 1.
- Snapshot version 1.
- Not modeled: Radiation outside the Wien regime; Broad bands; How the constrained states are prepared; Walls, mirrors, adiabatic compression, or any mechanism that changes volume; Any interpretation of E/(hν) as a count of particles.
Within the regime where Wien's law holds, how does the entropy of monochromatic radiation depend on the volume it occupies, and what had to be fixed to get a definite answer?
Fixed energy E = 9.055615 × 10−9 J at ν = 6.00 × 1014 Hz in a 1.00 × 1012 Hz band.
ΔS = 0.000000 J/K (closed form); 0.000000 J/K (numerical S(V) − S(V₀)); coefficient E/(βν) = 3.144807 × 10−13 J/K; E/(hν) = 2.277774 × 1010 (never a count of particles).
| Reference (V₀) | Compared (V) | |
|---|---|---|
| Temperature T | 3000.000 K | 3000.000 K |
| x = βν / T | 9.598486 | 9.598486 |
| Pointwise deviation e−x | 6.783135 × 10−5 | 6.783135 × 10−5 |
| Spectral entropy density sν | 3.333019 × 10−21 J/(m³ Hz K) | 3.333019 × 10−21 J/(m³ Hz K) |
Not modeled: Radiation outside the Wien regime; Broad bands; How the constrained states are prepared; Walls, mirrors, adiabatic compression, or any mechanism that changes volume; Any interpretation of E/(hν) as a count of particles.
The same action without dragging, color, or a canvas
Every action here is typed text entry and a text result: choose half, same, or double, or type any volume ratio, and read the entropy-change sentence and the state table above. No control depends on color, drag gestures, or a canvas.
When faint light of one colour is given more room at the same energy, its entropy rises by the same logarithmic law as an ideal gas that expands. That shared law is the clue the paper follows next.
Section 3 turns a radiation law into an entropy: at fixed energy, the entropy S = v∫φ(ρ, ν)dν is a maximum, and with dS = dE/T this gives ∂φ/∂ρ = 1/T, where φ vanishes when the density ρ is zero. Section 4 takes Wien's law, ρ = αν³e−βν/T, which experiment had confirmed for large ν/T, solves it for 1/T and integrates. For radiation of energy E in a narrow band from ν to ν + dν filling a volume v, the dependence on volume is S − S0 = (E/βν) ln(v/v0). The instrument holds E, ν and dν fixed, lets you choose the volume ratio, and refuses a state dense enough for Wien's law to fail.
Start from Wien's law, ρ = αν³e−βν/T, where ρ is the energy per unit volume and per unit of frequency. Taking the natural logarithm of both sides gives ln(ρ/αν³) = −βν/T, so 1/T = −(1/βν) ln(ρ/αν³). Section 3 showed that ∂φ/∂ρ = 1/T, so φ is found by integrating 1/T with respect to ρ from φ = 0 at ρ = 0; the result is φ = −(ρ/βν){ln(ρ/αν³) − 1}. Now put the energy E, in the band dν, inside a volume v. Then ρ = E/(v dν), and the entropy is S = vφ dν = −(E/βν){ln(E/(vαν³dν)) − 1}. Only one part of this depends on v: −(E/βν) ln(1/v), which is (E/βν) ln v. Subtracting the same expression at the volume v0 leaves S − S0 = (E/βν) ln(v/v0). At the instrument's defaults (ν = 6 × 1014 Hz, T0 = 3000 K, one litre, a band 1012 Hz wide) the snapshot has E = 9.06 × 10−9 J and E/(βν) = 3.14 × 10−13 J/K. Doubling the volume multiplies that coefficient by ln 2, about 0.693, so the entropy rises by 2.18 × 10−13 J/K. The same energy is now spread thinner, so ρ halves and βν/T rises by exactly ln 2, from 9.60 to 10.29: the radiation is colder, 2798 K, and further inside the regime where Wien's law holds.
Einstein printed lg for the natural logarithm and used β for the constant in Wien's exponent, so his coefficient E/βν is, in modern notation, kBE/(hν), since β = h/kB. He said plainly that Wien's law is not exactly valid and that his results hold only within certain limits. The comparison with an ideal gas or a dilute solution is his own sentence at the end of Section 4; reading the coefficient in terms of independent energy quanta of size Rβν/N comes only in Section 6, after the probability argument of Section 5. Planck's formula of 1901 was already known, and Einstein works in its Wien limit on purpose.
The physical argument
Wien's variational argument (§3)
Einstein attributes the entropy argument to Wien and uses it to fix the temperature dependence of the spectral entropy density . Maximizing at fixed energy, together with , gives:
Section 3 closes with the condition that fixes the integration constant: the entropy density vanishes when the radiation density is zero.
The dilute, narrow-band limit (§4)
Section 4 restricts to dilute monochromatic radiation obeying Wien's law, inverts it for the temperature, integrates the entropy density using the zero-density condition above, and integrates over a narrow band. The result is that the entropy of radiation of energy in volume depends on volume exactly the way the entropy of an ideal gas or a dilute solution does:
Only after this volume law is established does identifying give the modern form . The coefficient emerges from an entropy calculation; the workbench above never rounds it to an integer or calls it a count of particles.
Why the fixing condition matters
Had the integration constant been left unfixed rather than set to zero by the boundary condition, it would contribute an extra term proportional to that does not cancel and does not reproduce the volume law. The workbench's teaching panel above makes this concrete with an illustrative nonzero .
Epistemic boundary: a regime-limited approximation
Wien's law is an admitted approximation to the true (Planck) spectrum, accurate only where the radiation is dilute relative to the frequency and temperature in question. A state dense enough that Wien's law fails is not silently computed with a wrong answer; the workbench refuses it and states why, rather than presenting a number outside the regime the argument actually covers.
Embed this laboratory on another page. It starts from its worked defaults, not your current settings.