The Wilhelmina Ψ Field: A ContinuumTheory of Phase Noise in CoupledResonator Networks

Abstract

Phase noise limits performance in radar, 5G communications, satellite navigation, and precision timing systems. Existing theory treats phase noise at a single isolated oscillator (Leeson, 1966) or between two coupled oscillators (Adler, 1946; Kurokawa, 1973). No continuous field description exists for how phase noise power spectral density propagates across a network of many coupled resonators. This paper proposes the Wilhelmina Ψ field: a scalar field Ψ(x, f, t) representing phase noise power spectral density as a function of spatial position, offset frequency, and time. The governing reaction-diffusion partial differential equation is derived from first principles — from the Langevin stochastic oscillator equation, through the Fokker-Planck equation for phase statistics, to the continuum limit of the Adler coupling equation across N nodes. Key contributions: Wilhelmina diffusivity (D_Ψ): A new composite parameter quantifying how fast phase noise power propagates through a coupling network, with explicit circuit-level formula. Noise penetration depth (λ_Ψ): Distance over which phase noise decays to 37% — at typical coupling, λ_Ψ ≈ 18 μm, matching empirical chip keep-out zones with a first-principles basis. Backward compatible: Reduces to Leeson (1 node) and Adler (2 nodes). Closed-form solutions for linear chains, rings, and 2D arrays. Three validation protocols — spectrum analyser, SPICE, and Python Crank-Nicolson solver. Repository contains full paper, pedagogical guidebook, LTspice protocol, Python validation code, generated figures, and pre-computed simulation results.