Annus Mirabilis · Interactive critical edition in preparation
Radiation entropy workbench
Give faint light of one colour more room at the same energy and see its entropy rise.
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 explanation
Full explanation
Where Wien's law holds, the entropy of monochromatic radiation grows with its volume by the same logarithmic law as an ideal gas that expands. The lab keeps the energy and the frequency band fixed so the comparison is fair.
Show every step of the investigation
Halve or double the volume at fixed energy and band and read the entropy change. Then check where the result needs Wien's regime, and which constant the comparison leaves undetermined.
An explanatory model, not an observation of nature. This embed starts from the laboratory’s worked defaults, not a saved run. Presentation options change the surrounding guide, never the numerical inputs.