Spectral Entropy

📐 Definition

For spectral energy weights $p_k$ normalized from $\hat{u}(k)$,

$$p_k = \frac{|\hat{u}(k)|^2}{\sum_{j} |\hat{u}(j)|^2}, \qquad H = -\sum_{k} p_k \log p_k$$

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Domain and Codomain

Requires nonnegative energy spectrum $|\hat{u}(k)|^2$ with finite sum; $p_k$ forms a probability distribution. Entropy satisfies $0 \le H \le \log N$ for $N$ discrete modes (or extends to integrals in the continuous case).

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⚙️ Key Properties

Invariant to global scaling of $\hat{u}$. Minimal entropy (zero) occurs when energy is concentrated in one mode; maximal entropy $\log N$ when energy is uniform over $N$ modes.

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After ideal low-pass filtering, entropy typically decreases as energy concentrates at low $k$. For white spectra with $N$ equal modes, $H = \log N$.

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🎯 Special Cases and Limits

• Single-mode spectrum: $H=0$.
• Uniform energy across $N$ modes: $H=\log N$.

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🔗 Related Functions

Spectral bandwidth measures dispersion of energy in $k$-space; spectral filtering and phase modulation alter $p_k$ and thus entropy.

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Usage in Oakfield

Oakfield computes spectral entropy as a built-in diagnostic:

• SimFieldStats.spectral_entropy is computed by taking an FFT of the (flattened) field, forming normalized power $p_k$, then evaluating $-\sum p_k\log p_k$ with a log(n) normalization.
• Used to detect mode concentration (low entropy) vs broadband/noisy behavior (high entropy), especially when using spectral operators or filtering.

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Historical Foundations

📜 Shannon Entropy

Spectral entropy applies Shannon’s entropy to normalized spectral energy, treating the spectrum as a probability distribution over modes.

🌍 Modern Perspective

It is a common scalar diagnostic for concentration vs. spread in modal representations, especially in turbulence and wave simulations.

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📚 References

• Cover & Thomas, Elements of Information Theory
• Bracewell, The Fourier Transform and Its Applications