Blackbody Thermodynamics

An interesting (and unexpected!) application of the foundation of thermodynamics exposed earlier in this blog, is Planck’s law for the black body radiation. This is the plot of the radiation of a black body, where the horizontal axis is the frequency, and the three curves correspond to three different temperatures.

The curve is similar if one replaces the frequency by the wave length. Here is a curve of sunlight with and without atmospheric absorption. Surprisingly, the one with no atmospheric absorption follows the black-body prediction quite closely!

One can use the black-body model to understand at which freqencies (or wave-lengths) the earth radiates energy.

Thermodynamical Framework

Recall the modeling and notations of the foundation of thermodynamics post.

Suppose first that we found a space of microstates \(X\) with a corresponding measure \(δx\), an energy function \(E \colon X \to V\), where \(V\) is a vector space and an associated free energy function \(A \colon V^\ast \to \RR\) defined by \[ A(θ) = \log \Bigl(\int \exp(\langle θ,E(x)\rangle) δx\Bigr) \] with a corresponding mean energy \(U \colon V^{\ast} \to V\) at a given temperature \(θ \in V^{\ast}\) given by \[ U(θ) = \dd A \]

Step 1: Parameterized Thermodynamical System

It turns out that we can immediately create a system depending on an external parameter \(k \in \RR^d\) as follows:

This is a new system, with a scalar energy. We’ll denote the corresponding temperature by \(τ \in \RR\).

From the definition of \(A\), one obtains \[ A_k(τ) = A(\| k \| τ) \] The mean energy at a given temperature is thus given by \[ U_k(τ) = {A’_k}(τ) \]

So we finally have

\[U_k(τ) = \| k \| U(τ\|k\|) \]

Step 2: Family of Systems

The next postulate is that we have a family of systems:

Step 3: Spherical Integration (& Stefan–Boltzmann Law)

We can already find out how the total energy must depend on the temperature. Recall that we have \(U_k(τ) = \| k \| U(τ\|k\|) \).

We take advantage of the fact that \(U_k\) only depends on the norm \(\|k\|\). ket us define \(\nu := \|k\|\) and use a spherical change of coordinates to compute \begin{align} E(τ) &= \int_{\RR^d} U_k(τ) \dd k \\ &= \int_{\RR^d} \|k\|U(\|k\|τ) \dd k \\ &= \int_{0}^{∞} S_d \nu^{d} U(\nu τ) \dd \nu \\ \end{align} where \(S_d\) is the surface of a sphere in \(d\) dimensions.

Now, with the change of variable \(v := \nu τ\), we get \begin{align} E(τ)&= τ^{-(d+1)} \int_{0}^{∞} S_d v^d U(v) \dd v \\ &= τ^{-(d+1)} C_d \end{align} and \(C_d\) is a constant namely the integral \( C_d = ∫_{0}^{∞} S_d v^d U(v) \dd v \).

What we have found is the Stefan–Boltzmann law:

\[ E(τ) \propto τ^{-(d+1)} \]

Step 4: A “Quantum” Microstate Space

It remains to find an interesting example of microstate space \(X\) and its associated measure \(δx\).

It turns out that one such example is as follows.

Note that when this space was chosen, it was very surprising: why would the space of microstates be discrete? At the time, it made little sense. However, the explanation lies in quantum mechanics: the microstates are quantized. Nevertheless, here, we just have to find whatever space \(X\) which leads to a satisfactory distribution \(E(θ)\).

We use the identity \begin{align} \sum_{k=0}^{∞} \exp(kθ) = -\frac{1}{1-\exp(θ)} \end{align} since the sum is a geometric sum. This is only defined for \[θ < 0\]

We can then use that to compute \[ A(θ) = -\log(1- \exp(\theta)) \]

From this, we obtain \[ U(θ) = \frac{\exp(θ)}{1-\exp(θ)} \] So

\[ U(θ) = \frac{1}{\exp(-θ) -1} \]

Step 5

Recall the total energy \(E(τ) = \int_{0}^{∞} S_d \nu^d U(\nu τ) \dd \nu\), so \[ E(τ) = \int_{0}^{∞} S_d \nu^{d} \frac{1}{\exp(-\nu τ)-1} \dd \nu \]

The term inside the integral is Planck’s law! In other words, we have \[ E(τ) = \int B(\nu,τ) \dd \nu \] with

\[ B(\nu,τ) = S_d\frac{\nu^d}{\exp(-\nu τ)-1} \]

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Edit 2025-09: Rewrite with a computation of the mode density