Phase Coherence Measures

📐 Definition

For angles $\{\theta_j\}_{j=1}^{N}$, the mean phase coherence is

$$R = \left|\frac{1}{N} \sum_{j=1}^{N} e^{i\theta_j}\right|$$

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$R \in [0,1]$ measures alignment: $R=1$ indicates identical phases; $R \approx 0$ indicates phase dispersion.

Domain & Codomain

Domain: phase samples or complex unit vectors $e^{i\theta_j}$; Codomain: scalar coherence in $[0,1]$.

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⚙️ Key Properties

Shift Invariance

Replacing $\theta_j$ by $\theta_j + \phi$ leaves $R$ unchanged; only relative phase differences matter.

Circular Mean Interpretation

$R$ equals the magnitude of the circular mean of unit phasors: $R = \left|\frac{1}{N}\sum_j e^{i\theta_j}\right|$. Larger $R$ indicates tighter angular concentration.

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🎯 Special Cases and Limits

For two phases,

$$R = \left|\tfrac{1}{2}(e^{i\theta_1} + e^{i\theta_2})\right| = \sqrt{\tfrac{1+\cos(\theta_1-\theta_2)}{2}} = \left|\cos\!\left(\tfrac{\theta_1-\theta_2}{2}\right)\right|$$

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For uniformly distributed phases, $R$ tends to $0$ as $N \to \infty$.

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🔗 Related Functions

Complex Phase Provides phase samples for coherence calculation

Spectral Entropy Alternative dispersion measure

Complex Exponential Fundamental link to phasors and order parameters

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Usage in Oakfield

Applications in Oakfield

Oakfield computes phase-coherence-style order parameters as part of its runtime field diagnostics (SimFieldStats):

• Unweighted coherence phase_coherence is computed as $\left|\frac{1}{N}\sum z/|z|\right|$ over samples that pass a magnitude gate.
• Weighted coherence phase_coherence_weighted replaces the mean by an amplitude-weighted sum (weight $|z|$), improving robustness when many samples are near zero.
• Optional deramping reports phase_coherence_k0, computed after removing the dominant spectral bin’s linear phase ramp.
• Time smoothing + lock detection track phase_coherence_ema with an exponential moving average and a hysteresis lock flag, configurable via context-level phase-coherence settings.

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Historical Foundations

1. 📜 Synchronization Order Parameters

   Coherence measures of the form $R=\left|\frac{1}{N}\sum_j e^{i\theta_j}\right|$ are standard in circular statistics and became especially prominent through the Kuramoto model as an order parameter for collective synchronization.

2. 🌍 Modern Perspective

   In numerical simulations, coherence is used as a compact scalar diagnostic for phase organization across space, time, or modal decompositions.

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📚 References

• Kuramoto, Chemical Oscillations, Waves, and Turbulence
• Mardia & Jupp, Directional Statistics