Read · Light quanta: from entropy to an energy scale

§4 · dilute radiation and the volume law

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A derivation, to an approximation

The constant cannot simply be dropped

Why does the zero-radiation boundary condition matter?

The 2 printed paragraphs this passage explains

§4, paragraph 1: Wien's radiation law is known not to hold exactly, but experiment confirmed it very well when the frequency is large compared with the temperature. Einstein builds on it while keeping in mind that his results hold only within those limits.
§4, paragraph 2: Putting Wien's law into the relation from section 3 gives the entropy density. For radiation of energy E in a narrow band of frequencies, the entropy changes by E/(βν) times the logarithm of v/v₀ when its volume changes from v₀ to v.

u_\nu = A\,\nu^{3}\,\exp\left(-\frac{B\,\nu}{T}\right)

Wien spectral energy density is A times frequency cubed times the exponential of minus B frequency over temperature.

Divide by $A\nu^3$ and take the natural logarithm. Its argument is dimensionless. Solving for inverse temperature turns the entropy derivative from §3 into an explicit function of density.

\frac{1}{T} = -\frac{1}{B\,\nu}\,\ln\left(\frac{u_\nu}{A\,\nu^{3}}\right)

Inverse temperature is minus one over B frequency times the natural logarithm of density over A frequency cubed.

s_\nu = -\frac{u_\nu}{B\,\nu}\,\left(\ln\left(\frac{u_\nu}{A\,\nu^{3}}\right) - 1\right) + C\left(\nu\right)

The integrated entropy density is minus density over B frequency times the bracketed logarithm minus one, plus a frequency-dependent constant.

As positive density approaches zero, $\rho\ln\rho$ tends to zero. The remaining limit is $C(\nu)$ . The boundary condition $s_\nu \to 0$ therefore sets $C(\nu) = 0$ . Without this condition, the volume comparison in the next step has an extra term.

Open the foundation: Logarithms: turning products into sums

Show every step here: The constant cannot simply be dropped

Letters: the paper writes Wien's law as $\rho=\alpha\nu^3e^{-\beta\nu/T}$ , with the α and β of §2 (A and B here), and writes lg for the natural logarithm (ln here). The paper notes that observations show this law is not exactly valid, but that for large values of ν/T it was confirmed very completely by experiment. It bases its calculation on the law and keeps in mind that the results hold only within certain limits.

u_\nu = A\,\nu^{3}\,\exp\left(-\frac{B\,\nu}{T}\right)

Wien spectral energy density is A times frequency cubed times the exponential of minus B frequency over temperature.

Fix ν. Divide both sides by $A\nu^3$ : $\frac{\rho_\nu}{A\nu^3}=\exp(-B\nu/T)$ . The left side has no units, since A carries the units that make $A\nu^3$ a spectral density.
Take the natural logarithm of both sides: $\ln\frac{\rho_\nu}{A\nu^3}=-\frac{B\nu}{T}$ .
Divide both sides by $-B\nu$ to leave 1/T alone. This is the paper's first result in §4, printed with lg.

\frac{1}{T} = -\frac{1}{B\,\nu}\,\ln\left(\frac{u_\nu}{A\,\nu^{3}}\right)

Inverse temperature is minus one over B frequency times the natural logarithm of density over A frequency cubed.

Check the sign. A density that obeys Wien's law is smaller than $A\nu^3$ , so the ratio is below 1 and its logarithm is negative; the minus sign in front makes 1/T positive, as it must be.
§3 found that the entropy derivative is 1/T. Substitute: $\frac{\partial s_\nu}{\partial\rho_\nu}=-\frac{1}{B\nu}\ln\frac{\rho_\nu}{A\nu^3}$ .
Integrate with respect to $\rho_\nu$ at fixed ν. Write $a=A\nu^3$ for short. The antiderivative of $\ln(\rho/a)$ is $\rho\ln(\rho/a)-\rho$ ; differentiating it gives back $\ln(\rho/a)+1-1$ .
Multiply by $-1/(B\nu)$ and add a constant of integration. Because ν was held fixed, the constant may depend on ν, so it is written C(ν).

s_\nu = -\frac{u_\nu}{B\,\nu}\,\left(\ln\left(\frac{u_\nu}{A\,\nu^{3}}\right) - 1\right) + C\left(\nu\right)

The integrated entropy density is minus density over B frequency times the bracketed logarithm minus one, plus a frequency-dependent constant.

Now the boundary condition. As $\rho_\nu$ falls toward zero, $\rho\ln\rho$ also falls toward zero, because the logarithm grows more slowly than ρ shrinks, and so does ρ itself. What remains in the limit is C(ν).
Worked limit, in units of a: at $\rho/a=10^{-3}$ , $\rho\ln(\rho/a)$ is -0.0069; at $\rho/a=10^{-6}$ it is $-1.38\times10^{-5}$ . The density-dependent terms shrink toward zero.
The paper requires the entropy to vanish when there is no radiation (φ = 0 at ρ = 0, end of §3). So C(ν) = 0, and the paper's printed result is the bracket without any constant: $\varphi(\rho,\nu)=-\frac{\rho}{\beta\nu}\left\{\lg\frac{\rho}{\alpha\nu^3}-1\right\}$ .
The constant was fixed by a stated condition; it was not dropped for looking small. Kept, it would add a term proportional to the volume to the comparison in the next step.

A tool for this step

Logarithms: turning products into sums

Why does entropy need a logarithm?

Adding is easier than multiplying, and a logarithm turns one into the other. The natural logarithm of 2 is about 0.693, and of 4 about 1.386. Add them and you get 2.079, which is the natural logarithm of 8, the product of 2 and 4.

The natural logarithm ln x answers one question: to what power must e, about 2.718, be raised to give x? Raised to 0.693, e gives 2, and raised to 1.386 it gives 4. Multiplying two powers of e adds the powers, so raised to 2.079 it gives 8. Written for any numbers, the logarithm of a product is the sum of the logarithms, and a power becomes a multiple:

\ln\left(A\,B\right) = \ln\left(A\right) + \ln\left(B\right)

\ln\left(f^{n}\right) = n\,\ln\left(f\right)

The log of A times B is log A plus log B, and the log of f to the power n is n times log f.

Entropy needs exactly this. In §5 of the light-quanta paper, Einstein takes two independent systems. Their entropies add, S = S₁ + S₂, while the probabilities of their states multiply, W = W₁W₂. If entropy is some function φ of probability, then φ(W₁W₂) = φ(W₁) + φ(W₂), and he concludes that φ is a logarithm:

S - S_{0} = \frac{R}{N}\,\ln\left(W\right)

S minus S nought equals R over N times the natural log of W.

R is the gas constant and N the number of molecules in a mole, so R/N is what is now called Boltzmann's constant. The step from φ(W₁W₂) = φ(W₁) + φ(W₂) to the logarithm needs φ to be a smooth function, which Einstein takes for granted.

Einstein prints lg for the natural logarithm, as German physics did in 1905. Today lg usually means the base-10 logarithm and ln the natural one. So a printed lg 2 means ln 2, about 0.693, not log₁₀ 2, about 0.301.

Worked example: Ten molecules found in half the volume

In §5 Einstein asks how probable it is that n molecules, each moving independently through a volume v₀, are all found at one moment in a part v of it. Each molecule is in that part with probability v/v₀, and independent probabilities multiply:

W = \left(\frac{v}{v_{0}}\right)^{n}

W equals v over v nought, to the power n.

Take n = 10 molecules and v = v₀/2. Then W = (1/2)¹⁰ = 1/1024, about 0.000977.
The logarithm turns the tenth power into ten times: ln W = 10 × ln(1/2) = 10 × (−0.693) = −6.93.
So the entropy of the state with every molecule in one half is lower by 6.93 × R/N: 6.93 times Boltzmann's constant.

For a mole of gas, n = N, and the same formula gives S − S₀ = R ln(v/v₀). That is the entropy change thermodynamics gives for an ideal gas compressed from v₀ to v at constant temperature.

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Try it: turning multiplication into addition

In §5 of the light-quanta paper, Einstein takes two independent systems. The probabilities of their states multiply, W = W₁ · W₂, while their entropies add, S = S₁ + S₂. A logarithm turns the one into the other, which is why he concludes that entropy is a constant times ln W, plus a constant. The table checks the rule on three pairs of numbers.

Notation display toggle:

Switches the table between the symbol printed in 1905, lg, and the modern ln.

1905 historical notation vs Modern ISO standard

In 1905 German scientific printing (including Annalen der Physik), the symbol lg denoted the natural logarithm (base e).

In modern ISO 80000-2 notation, ln denotes the natural logarithm, while lg is reserved for the common base-10 logarithm (log₁₀).

1905 printed “lg 2”: 0.693147, the natural logarithm ln 2.

Modern ISO “lg 2” (log₁₀ 2): 0.301030, the common base-10 logarithm.

ln(W₁ · W₂) = ln(W₁) + ln(W₂), checked on three pairs
W₁ and W₂	ln(W₁) + ln(W₂)	ln(W₁ · W₂)
2 and 4 product 8	0.693147 + 1.386294 = 2.079442	ln 8 = 2.079442
3 and 5 product 15	1.098612 + 1.609438 = 2.708050	ln 15 = 2.708050
10 and 10 product 100	2.302585 + 2.302585 = 4.605170	ln 100 = 4.605170

What it shows, in words

For each of three pairs of numbers, the logarithm of the product equals the sum of the two logarithms. That is the property entropy needs: the probabilities of independent systems multiply and their entropies add. In the 1905 papers the logarithm is printed lg and means the natural logarithm, ln.

Where this lesson stops

A logarithm turns products into sums, which is why it connects probabilities that multiply to entropies that add. In the 1905 papers, lg means the natural logarithm.

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Integrate for the entropy density

s_\nu = -\frac{u_\nu}{B\,\nu}\,\left(\ln\left(\frac{u_\nu}{A\,\nu^{3}}\right) - 1\right) + C\left(\nu\right)

is minus over times , times the logarithm of density over A nu cubed minus one, plus .

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The integrated entropy density is minus density over B frequency times the bracketed logarithm minus one, plus a frequency-dependent constant.

Integrating 1/T with respect to the density gives the entropy density up to a constant C(nu) that may depend on frequency. The constant cannot simply be dropped; the next step fixes it by a boundary condition.

Model assumptions and every term’s meaning

The Wien regime: the density follows A nu cubed times exp(−B nu / T).
Entropy density and energy density are compared at one frequency.
These are modern teaching equations in SI notation; the paper writes L for c, R/N for Boltzmann's constant, and β for B.

B times nu: A temperature. Read the prerequisite
Logarithm minus one: The minus one is what integrating the logarithm adds. Read the prerequisite
nu cubed: From Wien's law. Read the prerequisite
Its logarithm: From inverting Wien's exponential. Read the prerequisite
Density over A nu cubed: A pure number, so its logarithm is defined. Read the prerequisite
Density over B nu: Carries the entropy density's units. Read the prerequisite
What it asserts: The entropy density of Wien-regime radiation as a function of its energy density, up to a constant the next step fixes. Read the prerequisite
The integrated term: Minus density over B nu times the bracket. Read the prerequisite
Plus the constant: Integration fixes the entropy density only up to this. Read the prerequisite
A nu cubed: Wien's prefactor at this frequency. Read the prerequisite
Wien's alpha: The constant multiplying Wien's law. Read the prerequisite
Wien's beta: The constant in Wien's exponent; later identified as h divided by Boltzmann's constant. It is not the β of the relativity paper. Read the prerequisite
The integration constant: Whatever integration leaves; the boundary condition that entropy density vanishes with energy density sets it to zero. Read the prerequisite
Spectral energy density: Radiation energy per cubic metre per unit of frequency. Read the prerequisite
Spectral entropy density: Entropy per cubic metre per unit of frequency; the paper writes it phi. Read the prerequisite
Frequency: The frequency of the radiation. Read the prerequisite

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Wien's law solved for the temperature

\frac{1}{T} = -\frac{1}{B\,\nu}\,\ln\left(\frac{u_\nu}{A\,\nu^{3}}\right)

One over is fixed by , through a logarithm and .

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One over the temperature equals minus one over B times the frequency, times the natural logarithm of the density over A times the frequency cubed.

Taking the logarithm of Wien's law and solving for one over the temperature expresses the temperature through the density. With one over T equal to the entropy's slope, this is what gets integrated.

Model assumptions and every term’s meaning

Dilute radiation: the density is low enough, relative to temperature, that Wien's law describes the spectrum.
These are modern teaching equations in SI notation, with A and B for the paper's alpha and beta; they are not a transcription of the printed paper.

Temperature: Its reciprocal, which the entropy's slope with respect to energy equals. Read the prerequisite
Spectral constant B: Divides out of the exponent when the logarithm is taken. Read the prerequisite
Frequency: Held fixed: each frequency band is treated on its own. Read the prerequisite
Spectral energy density: The variable the entropy will be integrated over. Read the prerequisite
Spectral constant A: The paper's alpha, inside the logarithm. Read the prerequisite
Frequency: Cubed, as in Wien's law. Read the prerequisite
The law turned inside out: The same law with the logarithm taken, so the temperature is given by the density rather than the other way round. Read the prerequisite

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Wien's spectrum

u_\nu = A\,\nu^{3}\,\exp\left(-\frac{B\,\nu}{T}\right)

is times cubed, damped by an exponential in times the frequency over .

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The spectral energy density equals A times the frequency cubed times the exponential of minus B times the frequency over the temperature.

Wien's law for the energy density on the frequency axis. It fits the measured spectrum where B times the frequency is large compared with the temperature, and fails where it is small.

Model assumptions and every term’s meaning

Dilute radiation: the density is low enough, relative to temperature, that Wien's law describes the spectrum.
These are modern teaching equations in SI notation, with A and B for the paper's alpha and beta; they are not a transcription of the printed paper.

Spectral energy density: Energy per unit volume per unit frequency interval. Read the prerequisite
Spectral constant A: The paper's alpha, fitted to measurement. Read the prerequisite
Frequency: Cubed in front, so the spectrum rises steeply before the exponential turns it down. Read the prerequisite
Spectral constant B: The paper's beta, in the exponent. Its product with the frequency over the temperature is a pure number. Read the prerequisite
Frequency: The same frequency, in the exponent. Read the prerequisite
Temperature: A hotter body moves the peak to higher frequency. Read the prerequisite
Wien's law: A measured law, taken as given in the dilute regime. The paper builds on it; it does not derive it. Read the prerequisite

Every term’s units were checked when this page was built. That checks the units, not the model. This equation is written for this edition in modern notation, not transcribed from the paper.

Modern qualifications

This explanation uses natural logarithms and editorial $A$ , $B$ notation. The source prints $\lg$ . The interpretation of $B$ as $h/k_B$ belongs to the later comparison of coefficients.

Assumptions and limits: The constant cannot simply be dropped

Assumed here

The narrow spectral region obeys the Wien approximation.
At zero radiation density, radiation entropy density vanishes.

What this does not establish

A positive inferred temperature alone is not a guarantee that the Wien approximation is accurate.
The spectral constants are not initially interpreted as particle energies.

Earlier step: A spectrum can determine an entropy derivative

Source context: German source · English · Interlinear gloss · Facsimile

A derivation, to an approximation

Compare two states, not a compression movie

What is held fixed when volume changes?

The 2 printed paragraphs this passage explains

§4, paragraph 2: Putting Wien's law into the relation from section 3 gives the entropy density. For radiation of energy E in a narrow band of frequencies, the entropy changes by E/(βν) times the logarithm of v/v₀ when its volume changes from v₀ to v.
§4, paragraph 3: So the entropy of dilute monochromatic radiation depends on its volume by the same law as the entropy of an ideal gas or a dilute solution. Einstein will interpret this with Boltzmann's principle, that the entropy of a system is a function of the probability of its state.

Within a narrow band, write $E = V\,\Delta\nu\,\rho_\nu$ and $S = V\,\Delta\nu\,s_\nu$ . Substitute $\rho_\nu = E/(V\,\Delta\nu)$ into the integrated entropy density. Comparing $V$ with $V_0$ at the same $E$ and band removes terms independent of volume.

S\left(V\right) - S\left(V_{0}\right) = \frac{E}{B\,\nu}\,\ln\left(\frac{V}{V_{0}}\right)

The entropy difference is E over B frequency times the natural logarithm of the volume ratio.

$E/(B\nu)$ has entropy units. Only after identifying $B = h/k_B$ can it be written $k_B$ times the dimensionless coefficient $E/(h\nu)$ . This distinction matters when interpreting a displayed coefficient as a count.

\Delta S_C = \Delta\nu\,C\left(\nu\right)\,\left(V - V_{0}\right)

An unfixed entropy-density constant would add bandwidth times C of frequency times the volume difference.

Doubling the accessible volume adds $(E/(B\nu))\ln 2$ under the stated constraints. Doubling it again adds the same amount. The radiation-entropy laboratory compares those constrained states and exposes the otherwise hidden constant.

Open the foundation: Partial derivatives and held-fixed quantities

Show every step here: Compare two states, not a compression movie

Letters: the paper's radiation has energy E in the narrow band between ν and ν + dν and fills a volume v; this explanation writes the band width Δν and the volume V. $S_0$ is the entropy when the same radiation fills the volume $v_0$ , written $V_0$ here.

In the band, the energy is volume times band width times density: $E=V\Delta\nu\,\rho_\nu$ , so $\rho_\nu=E/(V\Delta\nu)$ .
The entropy is volume times band width times entropy density: $S=V\Delta\nu\,s_\nu$ .
Insert the entropy density of the previous step, keeping C(ν) for now: $S=-\frac{E}{B\nu}\left[\ln\frac{E}{VA\nu^3\Delta\nu}-1\right]+V\Delta\nu\,C(\nu)$ . The factor $V\Delta\nu$ times $\rho_\nu/(B\nu)$ has become $E/(B\nu)$ .
Write the same expression with $V_0$ in place of V, at the same E, the same ν and the same band.
Subtract. Every term without V cancels. Inside the logarithms, $\ln\frac{E}{VA\nu^3\Delta\nu}-\ln\frac{E}{V_0A\nu^3\Delta\nu}=\ln\frac{V_0}{V}$ , and the minus sign in front turns it into $\ln\frac{V}{V_0}$ .

S\left(V\right) - S\left(V_{0}\right) = \frac{E}{B\,\nu}\,\ln\left(\frac{V}{V_{0}}\right)

The entropy difference is E over B frequency times the natural logarithm of the volume ratio.

The C(ν) term does not cancel. It leaves the piece below, proportional to the difference of the volumes. That is why the constant had to be fixed, and the zero-radiation condition fixed it at zero.

\Delta S_C = \Delta\nu\,C\left(\nu\right)\,\left(V - V_{0}\right)

An unfixed entropy-density constant would add bandwidth times C of frequency times the volume difference.

Worked numbers: halve the volume, $V/V_0=1/2$ , and the entropy falls by $(E/B\nu)\ln2$ , about 0.693 times $E/(B\nu)$ . Halve it again and it falls by the same amount again. Only the ratio of the volumes matters, not their size.
Units: B is in seconds times kelvin, so Bν is in kelvin and $E/(B\nu)$ is energy per kelvin, the unit of entropy. Only after reading B as $h/k_B$ , a later identification, does $E/(B\nu)$ become $k_B$ times the pure number $E/(h\nu)$ .
The paper notes that the entropy of monochromatic radiation of low enough density varies with volume by the same law as the entropy of an ideal gas or a dilute solution. §5 asks where that law comes from for a gas.
Check the conditions at both ends: the same energy, the same narrow band, and a density low enough for Wien's law in both volumes. This compares two states. A mirror that moves while the radiation is inside changes frequencies and energy, which is a different comparison.

A tool for this step

Partial derivatives and held-fixed quantities

When a quantity depends on two things, what does its rate of change mean?

The concentration of particles in a tube depends on where you look, x, and when you look, t. Ask how fast the concentration changes and there are two different answers: how it changes as you move along the tube at one instant, and how it changes at one place as time passes.

A partial derivative picks one of them. ∂f/∂x is the rate of change along the tube with the time t held fixed, like comparing points in one photograph. ∂f/∂t is the rate of change in time at a fixed position x, like watching one spot under the microscope. The curly ∂ says that something else is being held fixed, and a subscript names it:

\left(\frac{\partial f}{\partial x}\right)_t \qquad \left(\frac{\partial f}{\partial t}\right)_x

The partial derivative of f with respect to x at fixed t, and the partial derivative of f with respect to t at fixed x.

Thermodynamics depends on the same care. How much a gas's volume changes as the pressure changes depends on whether its temperature T is held fixed, with heat flowing in and out, or its entropy S is held fixed, with the gas insulated. The two derivatives have different values, and ∂V/∂p written alone does not say which is meant:

\left(\frac{\partial V}{\partial p}\right)_T \quad \text{versus} \quad \left(\frac{\partial V}{\partial p}\right)_S

The partial derivative of V with respect to p at fixed temperature T, versus the same derivative at fixed entropy S.

In §3 of the light-quanta paper, radiation fills a fixed volume and φ is its entropy per unit volume and per unit interval of frequency, a function of the energy density ρ and of ν. Varying ρ with the frequency ν held fixed, Einstein finds that the derivative is 1/T:

\left(\frac{\partial \varphi}{\partial \rho}\right)_{\nu} = \frac{1}{T}

The partial derivative of phi with respect to rho, at fixed frequency nu, equals one over T.

In §6 of the relativity paper, Maxwell's equations are rewritten in the coordinates ξ, η, ζ, τ of a moving system. A derivative with respect to x taken with y, z and t held fixed then becomes a combination of a derivative with respect to ξ and one with respect to τ, because holding the resting time t fixed does not hold the moving time τ fixed.

Worked example: The same diffusion profile, differentiated two ways

Take the spreading profile of the Brownian paper's §4, written for a single particle:

f\left(x,\,t\right) = \frac{1}{\sqrt{4\,\pi\,D\,t}}\,\exp\left(-\frac{x^{2}}{4\,D\,t}\right)

f of x and t equals one over the square root of four pi D t, times e to the minus x squared over four D t.

Hold t fixed and differentiate with respect to x. Only the exponent depends on x, and the derivative of −x²/4Dt with respect to x is −x/2Dt:

\left(\frac{\partial f}{\partial x}\right)_{t} = -\frac{x}{2\,D\,t}\,f\left(x,\,t\right)

The partial derivative of f with respect to x at fixed t equals minus x over two D t, times f.

That is the slope of the profile in one snapshot: downhill to the right of the centre, uphill to the left. Now hold x fixed and differentiate with respect to t. Both the factor in front and the exponent depend on t:

\left(\frac{\partial f}{\partial t}\right)_{x} = \left(-\frac{1}{2\,t} + \frac{x^{2}}{4\,D\,t^{2}}\right)\,f\left(x,\,t\right)

The partial derivative of f with respect to t at fixed x equals minus one over two t, plus x squared over four D t squared, all times f.

That is what a probe fixed at one place records. Near the centre, where x² is less than 2Dt, the bracket is negative: the concentration falls as the cloud spreads away. Farther out it is positive: the concentration rises as the cloud arrives. Differentiate the first result once more with respect to x and multiply by D, and you get the second. That is the §4 equation ∂f/∂t = D ∂²f/∂x², a rate at one place equal to a curvature at one time.

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Try it: hold time fixed, or hold position fixed

In §4 of the Brownian paper, f(x, t) is the number of particles per unit volume at a place x along a tube, at a time t. “How fast does the concentration change?” then has two answers, and which one you mean depends on what you keep still: the moment, or the place.

Keep one of them still:

One moment: t fixed, ∂f/∂x

Freeze the moment and compare neighbouring places, as in a single photograph of the tube. ∂f/∂x says how steeply the concentration changes along it: particles per µm³ for each µm, which is particles per µm⁴.

Particles drift down that slope. The flux, the number crossing each square micrometre in a second, is J = −D ∂f/∂x: Fick’s first law.

One place: x fixed, ∂f/∂t

Keep your eye on one spot under the microscope and let time run. ∂f/∂t says how fast the count there rises or falls: particles per µm³ for each second.

Particles are neither made nor destroyed, so the count at a spot changes only by what flows in and out: ∂f/∂t = −∂J/∂x = D ∂²f/∂x². That is the diffusion equation of §4.

The same question in thermodynamics

Squeeze a gas and its pressure rises, but by how much depends on what else you keep fixed:

Three derivatives of pressure, and what each holds fixed
Held fixed	Derivative	What happens
Temperature T (isothermal)	(∂p / ∂V)_T	Heat flows in or out to keep the temperature constant.
Entropy S: no heat exchanged (adiabatic)	(∂p / ∂V)_S	The gas warms as it is squeezed, so its pressure climbs faster than at fixed temperature.
Volume V (isochoric)	(∂p / ∂T)_V	In a rigid vessel the pressure rises as the gas is heated.

What it shows, in words

∂f/∂x compares places at one moment; ∂f/∂t follows one place through time. They answer different questions and carry different units. In thermodynamics a subscript names what is held fixed, so (∂p/∂V)_T and (∂p/∂V)_S are different numbers for the same gas.

Where this lesson stops

A partial derivative is not defined until you have said which quantities are held fixed.

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What an unfixed constant would add

\Delta S_C = \Delta\nu\,C\left(\nu\right)\,\left(V - V_{0}\right)

would be times times minus .

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An unfixed entropy-density constant would add bandwidth times C of frequency times the volume difference.

If the entropy density's integration constant C(nu) were not set to zero, the comparison of two volumes would pick up this extra term, proportional to the volume difference rather than to the logarithm of the ratio.

Model assumptions and every term’s meaning

Radiation in a narrow band of width Δν around ν, compared in volumes V and V₀.
These are modern teaching equations in SI notation; the paper writes L for c, R/N for Boltzmann's constant, and β for B.

Band times constant times volume change: Linear in the volume difference: a different shape from the logarithm. Read the prerequisite
What it asserts: Why the constant cannot simply be dropped: the boundary condition that entropy density vanishes with energy density is what sets it to zero. Read the prerequisite
V minus V₀: A difference of volumes, not their ratio. Read the prerequisite
Width of the band: The width of the narrow frequency band. Read the prerequisite
The integration constant: C at this frequency: what integrating the entropy relation leaves undetermined. Read the prerequisite
The extra entropy: The term the constant would add to the comparison. Read the prerequisite
Frequency: The frequency of the radiation. Read the prerequisite
Volume: The volume the radiation fills. Read the prerequisite
Comparison volume: The volume the first state is compared with. Read the prerequisite

Every term’s units were checked when this page was built. That checks the units, not the model. This equation is written for this edition in modern notation, not transcribed from the paper.

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The entropy depends on the volume through a logarithm

S\left(V\right) - S\left(V_{0}\right) = \frac{E}{B\,\nu}\,\ln\left(\frac{V}{V_{0}}\right)

minus is over times times the logarithm of over .

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The entropy difference is E over B frequency times the natural logarithm of the volume ratio.

At the same energy and band, the radiation's entropy in volume V exceeds that in V₀ by E over Bν times the logarithm of their ratio. Doubling the volume adds (E/Bν) ln 2, and doubling it again adds the same.

Model assumptions and every term’s meaning

Radiation of fixed energy E in a narrow band of frequencies around ν, compared in two volumes V and V₀.
The Wien regime, with the integration constant of the entropy density set to zero.
These are modern teaching equations in SI notation; the paper writes L for c, R/N for Boltzmann's constant, and β for B.

B times nu: A temperature: B has units of seconds times kelvin. Read the prerequisite
E over B nu: An entropy, in joules per kelvin. Only once B is identified as h divided by Boltzmann's constant can it be read as Boltzmann's constant times a pure number. Read the prerequisite
The entropy difference: Only the difference is fixed; terms that do not depend on volume cancel. Read the prerequisite
Its logarithm: Doubling the volume adds ln 2, whatever the starting volume. Read the prerequisite
Coefficient times logarithm: The same form as for a gas of independent particles, which the next step compares. Read the prerequisite
What it asserts: How the radiation's entropy depends on its volume, at fixed energy and band. Read the prerequisite
Volume ratio: A pure number. Read the prerequisite
Wien's beta: The constant in Wien's exponent; later identified as h divided by Boltzmann's constant. It is not the β of the relativity paper. Read the prerequisite
Radiation energy: The fixed energy of the radiation in the band. Read the prerequisite
Entropy in volume V: The radiation's entropy when it fills volume V. Read the prerequisite
Entropy in volume V₀: The same radiation's entropy in the comparison volume. Read the prerequisite
Frequency: The frequency of the radiation. Read the prerequisite
Volume: The volume the radiation fills. Read the prerequisite
Comparison volume: The volume the first state is compared with. Read the prerequisite

Every term’s units were checked when this page was built. That checks the units, not the model. This equation is written for this edition in modern notation, not transcribed from the paper.

Modern qualifications

A modern pointwise comparison gives relative spectral-density error $\exp(-h\nu/(k_B T))$ for Wien versus Planck. That diagnostic is not, by itself, an error bound for the entire integrated entropy argument.

Assumptions and limits: Compare two states, not a compression movie

Assumed here

Energy E, central frequency ν, and bandwidth Δν are fixed in both states.
The entropy-density constant has been fixed by the zero-radiation condition.
Both states satisfy the dilute Wien-regime and narrow-band assumptions.

What this does not establish

A moving mirror generally changes frequency and energy, so it is not this held-fixed comparison.
The result is not valid merely because one of the two endpoint states is dilute.

Earlier step: The constant cannot simply be dropped

Source context: German source · English · Interlinear gloss · Facsimile

References

A. Einstein, Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt. Annalen der Physik (4), 17, 132–148 (1905).

A. Einstein, On the motion of particles suspended in liquids at rest required by the molecular-kinetic theory of heat. Annalen der Physik (4), 17, 549–560 (1905), §§1–5. Bibliographic pointer; this preview is not a source transcription or translation.

A. Einstein, Zur Elektrodynamik bewegter Körper. Annalen der Physik (4), 17, 891–921 (1905).

A. Einstein, Does the inertia of a body depend upon its energy content? Annalen der Physik (4), 18, 639–641 (1905). External 1923 Perrett–Jeffery translation, electronically transcribed by John Walker; its notation was modernized. A reference for this explanatory preview, not this edition’s translation or its pinned facsimile.

R. A. Millikan, A Direct Photoelectric Determination of Planck's "h". Physical Review (2), 7 (3), 355–388 (1916).

26 foundation readings sit behind this argument.

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