The Informational Equilibrium Restoration Law describes the progressive collapse of deviation-from-mean energy in a dynamic informational field. When diffused under a Laplacian operator, every component of the field converges toward its equilibrium value — the global mean.
Laplacian dynamics drive all informational states toward equilibrium.
This law governs stabilization, consensus processes, diffusion-driven averaging, and equilibrium formation across physical, informational, and cognitive systems.
The following experiment measures how equilibrium deviation energy evolves when the field undergoes Laplacian smoothing.
# Initial field with oscillatory + noisy structure A = sin(8πx) + 0.4 sin(20πx) + noise # Equilibrium energy: deviation from mean E = Σ (A - mean(A))² # Evolution: Laplacian smoothing A ← A + α ΔA # Result: equilibrium deviation energy strictly decreases.
The generated curve shows a smooth, monotonic collapse toward equilibrium, confirming the law experimentally. (Source computation: :contentReference[oaicite:1]{index=1})
E = Σ(A - mean(A))² is a Lyapunov functional.The energy decay follows:
E(t) = Σ (λᵢ² e^{-2 α |λᵢ| t})
where λᵢ are Laplacian eigenvalues.
This makes equilibrium restoration deeply connected to:
The Laplacian’s nullspace consists solely of constant vectors. Thus, only uniform fields remain unchanged under diffusion.
Because higher-frequency modes have more curvature, they collapse drastically faster than low-frequency ones.
Equilibrium restoration corresponds mathematically to variance decay in mean-reverting stochastic processes.
Submissions, analysis extensions, and alternative formulations can be provided via:
Submit via Email Submit via Zenodo