The Informational Hyperdiffusion Law describes the collapse of fourth-derivative energy in a field subjected to 4th-order diffusion. This form of smoothing is significantly stronger than ordinary Laplacian diffusion, producing extremely rapid attenuation of high-curvature structures.
Fourth-order diffusion drives accelerated collapse of high-derivative energy.
The law appears in geometric flows, beam equations, denoising algorithms, high-order PDE systems, and informational geometry models where curvature behaves as a higher-order structural quantity.
The following experiment measures the 4th-derivative energy
E = Σ (Δ²A)² as a field evolves under a hyper-Laplacian operator.
# Informational field with multi-frequency components
A = sin(12πx) + 0.5 sin(30πx) + noise
def lap(A): return roll(A,-1) - 2A + roll(A,1)
def hyper_lap(A): return lap(lap(A))
def hyper_energy(A):
return Σ (Δ²A)²
alpha = 0.00005
E = [hyper_energy(A)]
for step in range(300):
A = A - alpha * hyper_lap(A)
E.append(hyper_energy(A))
# Result: hyper_energy strictly decreases.
The recorded energy curve decreases smoothly across all iterations, confirming the expected hyperdiffusive collapse. (Source: :contentReference[oaicite:1]{index=1})
A ← A − α Δ²A.k⁴ for harmonic mode k.Fourth-order diffusion is strongly linked to:
Because hyperdiffusion evolves according to a gradient flow of the curvature-squared energy functional, the collapse is guaranteed to be monotonic.
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