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?

Read Section 3 of Einstein's 1905 paper →

LQ-04 · Radiation entropy workbench

The radiation entropy workbench

Ideal model, host calculation

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?

JavaScript is off. This is a complete worked example calculated when the site was built, at nu = 6.00e+14 Hz and T0 = 3000 K with the printed formulas below. Changing settings requires JavaScript.

Fixed energy E = 9.055615e-9 J at nu = 6.00e+14 Hz in a 1.00e+12 Hz band.

Delta S = 0.000000e+0 J/K (closed form); 0.000000e+0 J/K (numerical S(V) - S(V0)); coefficient E/(B nu) = 3.144807e-13 J/K; E/(h nu) = 2.277774e+10 (never a count of particles).

Constrained-state comparison at fixed E, nu, and dNu
	Reference (V0)	Compared (V)
Temperature T	3.000000e+3 K	3.000000e+3 K
x = B nu / T	9.598486e+0	9.598486e+0
Pointwise deviation e^-x	6.783135e-5	6.783135e-5
Spectral entropy density s_nu	3.333019e-21 J/(m^3 Hz K)	3.333019e-21 J/(m^3 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/(h*nu) as a count of particles.

Action contract: 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.

Show the code

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

\varphi

. Maximizing

S = v\int\varphi(\rho,\nu)\,d\nu

at fixed energy, together with

dS = dE/T

, gives:

\frac{\partial\varphi}{\partial\rho} = \frac{1}{T}

Section 3 closes with the condition that fixes the integration constant: the entropy density

\varphi

vanishes when the radiation density

\rho

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

E

in volume

v

depends on volume exactly the way the entropy of an ideal gas or a dilute solution does:

S - S_0 = \frac{E}{\beta\nu}\ln\frac{v}{v_0}\qquad(\text{printed "lg" is the natural logarithm})

Only after this volume law is established does identifying

\beta = h/k_B

give the modern form

\Delta S = k_B\,\frac{E}{h\nu}\ln\frac{V}{V_0}

. The coefficient

E/(h\nu)

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

C(\nu)

been left unfixed rather than set to zero by the boundary condition, it would contribute an extra term proportional to

(V - V_0)

that does not cancel and does not reproduce the volume law. The workbench's teaching panel above makes this concrete with an illustrative nonzero

C(\nu)

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.

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