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Thermodynamics of the blackbody radiation

1. What is Being Studied?

We look at blackbody radiation: electromagnetic waves (light) inside a cavity at temperature \(T\).

Goal: use Bose–Einstein statistics (the statistics for identical bosons) to describe the thermodynamics of this radiation.

There are two equivalent ways to think about the system:

  1. Oscillator picture (Planck's way)

  2. Each mode of the electromagnetic field (each possible frequency \(\omega_s\)) is treated as a harmonic oscillator with energies \((n_s + \tfrac{1}{2})\hbar \omega_s\) (where \(n_s = 0,1,2,\dots\)).

  3. For thermodynamics, the constant zero-point part \(\tfrac{1}{2}\hbar\omega_s\) doesn't matter, so we effectively use energies \(n_s \hbar\omega_s\).

  4. Photon gas picture (Bose & Einstein's way)

  5. The field is a gas of photons.

  6. Each photon has energy \(\hbar\omega_s\).
  7. Photons are identical bosons and any number of photons can occupy the same state.

The book shows that these two views give exactly the same results.

2. Average Energy of a Single Mode

Using Bose–Einstein statistics, the average number of photons in one mode of frequency \(\omega_s\) is

\[ \langle n_s \rangle = \frac{1}{e^{\hbar \omega_s / kT} - 1}. \]

Multiply by the photon energy \(\hbar\omega_s\) to get the average energy of that mode:

\[ \langle \varepsilon_s \rangle = \hbar\omega_s \langle n_s \rangle = \frac{\hbar\omega_s}{e^{\hbar\omega_s/kT} - 1}. \]

This is equation (1) on the first page.

Interpretation:

  • Low frequency (\(\hbar\omega \ll kT\)): the denominator ≈ \(\hbar\omega/kT\) so \(\langle \varepsilon_s \rangle \approx kT\) → classical equipartition.
  • High frequency: the exponential is large, so the average energy becomes very small.

3. How Many Modes Are There?

Next they find the number of electromagnetic modes per volume whose frequency lies between \(\omega\) and \(\omega + d\omega\).

This is the Rayleigh expression:

\[ \text{modes per unit volume between } \omega, \omega+d\omega = \frac{\omega^2 d\omega}{\pi^2 c^3}. \]

(This comes from counting standing waves in a box; the factors with \(4\pi\) and \((1/\lambda)^2\) are just that counting, rewritten with \(\omega = 2\pi c/\lambda\).)

4. Planck's Law from BE Statistics

Energy density in frequency range \((\omega, \omega + d\omega)\) = (number of modes) × (average energy per mode):

\[ u(\omega) d\omega = \frac{\hbar}{\pi^2 c^3} \frac{\omega^3 d\omega}{e^{\hbar\omega/kT} - 1}. \]

This is equation (3): Planck's distribution for blackbody radiation, now derived using Bose–Einstein statistics.

Meaning: This tells you how much energy per volume is carried by radiation with frequencies near \(\omega\).

5. Bose's and Einstein's Viewpoints

Bose

Bose didn't think in terms of oscillators; he thought directly about photons in energy levels:

  • He considered the probability that an energy level \(\varepsilon_s = \hbar\omega_s\) has \(n_s\) photons.
  • Using Boltzmann factors \(e^{-n_s\hbar\omega_s/kT}\) and summing over \(n_s\), he found the same average occupation:
\[ \langle n_s \rangle = \frac{1}{e^{\hbar\omega_s/kT} - 1}, \]

which leads again to the same average energy (equation (5)).

So Bose's counting of indistinguishable photons gives the same result as Planck's "oscillator energies" picture.

Einstein

Einstein went further:

  • He treated the photons and the energy levels together, emphasizing that photons are indistinguishable and that their total number is not fixed (photons can be created and destroyed by the walls).
  • In grand-canonical language: the chemical potential is zero, so the fugacity \(z = e^{\mu/kT} = 1\).
  • With this, he derived the very same BE distribution:
\[ \langle n_\varepsilon \rangle = \frac{1}{e^{\varepsilon/kT} - 1}. \]

That's equation (7) (with \(\varepsilon = \hbar\omega\)).

6. Rewriting in Dimensionless Form & Limiting Laws

They introduce a dimensionless variable

\[ x = \frac{\hbar\omega}{kT} \]

and rewrite the spectrum in terms of \(x\). Then the energy density becomes (equation (8)):

\[ u'(x) dx = \frac{x^3 dx}{e^x - 1}. \]

And \(u(x)\) itself is written as

\[ u(x) = \frac{\pi^2 \hbar^3 c^3}{(kT)^4} u(\omega) \]

(or similar — the details are just a scaling; see equation (9)).

Then they look at two limits:

  1. Long wavelengths / low frequencies (\(x \ll 1\)):

Expand \(e^x \approx 1 + x\). Then

[ u'(x) \approx x^2, ]

which corresponds to the classical Rayleigh–Jeans law.

  1. Short wavelengths / high frequencies (\(x \gg 1\)):

\(e^x \gg 1\), so

[ u'(x) \approx x^3 e^{-x}, ]

which is Wien's law.

The figure on page 202 shows the Planck curve with these two approximations.

7. Total Energy Density and Stefan–Boltzmann Law

To get the total energy density, integrate Planck's law over all frequencies:

\[ \frac{U}{V} = \int_0^\infty u(\omega) d\omega. \]

After changing variable to \(x\) and using a known integral \(\int_0^\infty \frac{x^3}{e^x - 1} dx = \frac{\pi^4}{15}\), one finds (equation (12)):

\[ \frac{U}{V} = \frac{\pi^2 k^4}{15\hbar^3 c^3} T^4. \]

So the energy density ∝ \(T^4\).

Now consider radiation escaping through a small hole in the cavity. The net energy flux (power per area) is

\[ \text{flux} = \frac{1}{4} c \frac{U}{V} = \sigma T^4 \]

with

\[ \sigma = \frac{\pi^2 k^4}{60\hbar^3 c^2} \]

(equation (14)). This is the Stefan–Boltzmann law.

8. Thermodynamics via the Grand Partition Function

They now treat the photon gas as a full thermodynamic system and compute its grand partition function:

\[ \ln \Theta(V,T) = \frac{PV}{kT} = -\sum_\varepsilon \ln(1 - e^{-\varepsilon/kT}). \]

(Equation (15); the sum is over all photon states.)

To turn the sum into an integral, they use the density of states in momentum space, with the relativistic relation for photons:

\[ \varepsilon = pc, \quad a(\varepsilon) d\varepsilon = \frac{8\pi V}{h^3 c^3} \varepsilon^2 d\varepsilon \]

(equation (16)). Using this and integrating by parts, they obtain

\[ \frac{PV}{kT} = \frac{8\pi V}{3h^3 c^3 kT} \int_0^\infty \frac{\varepsilon^3 d\varepsilon}{e^{\varepsilon/kT} - 1}. \]

Comparing with the earlier expression for \(U/V\), they find

\[ PV = \frac{1}{3} U. \]

That's equation (17).

Interpretation: Radiation pressure equals one third of the energy density — a standard result in radiation theory.

9. Other Thermodynamic Quantities

Once you know \(U\) and \(P\), you can get other quantities:

  • Helmholtz free energy \(A = -PV = -\tfrac{1}{3} U\). (Eq. 18)

  • Entropy:

\[ S = \frac{U - A}{T} = \frac{4U}{3T} \propto VT^3, \]

as in equation (19).

  • Heat capacity at constant volume:
\[ C_V = T \left(\frac{\partial S}{\partial T}\right)_V = 3S \]

(equation (20)).

For a reversible adiabatic change (no heat exchange), \(S\) is constant. Because \(S \propto VT^3\), this gives (equation (21)):

\[ VT^3 = \text{const}. \]

Since \(P \propto T^4\), you can combine these to get an equation for the adiabats:

\[ PV^{4/3} = \text{const} \]

(equation (22)), analogous to \(PV^\gamma = \text{const}\) with \(\gamma = 4/3\) for radiation.

10. Mean Number of Photons

Finally, they calculate the average total number of photons in the cavity:

\[ \bar{N} = \frac{V}{\pi^2 c^3} \int_0^\infty \frac{\omega^2 d\omega}{e^{\hbar\omega/kT} - 1} \propto VT^3 \]

(equation (23)).

So the number of photons grows like \(T^3\) and with the volume.

However, they warn that the fluctuations in \(N\) are very large (because photon number is not fixed and the compressibility is huge), so this average should be treated with care.

Big Picture Summary

  • Treating light in a cavity as a gas of photons (bosons) leads directly to the Planck spectrum.
  • The spectrum smoothly connects:
  • to Rayleigh–Jeans at low frequency,
  • to Wien's law at high frequency.
  • Integrating the spectrum gives:
  • Energy density \(U/V \propto T^4\),
  • Stefan–Boltzmann law for radiated power, \(\propto T^4\),
  • Radiation pressure \(P = U/3\),
  • Entropy \(S \propto VT^3\),
  • Adiabatic law \(PV^{4/3} = \text{const}\),
  • Mean photon number \(\bar{N} \propto VT^3\).