This is a 2D visualization of the recursive field Ψd(x,y)\Psi_d(x,y)Ψd(x,y) after evolving it over time.
Laplacian Operator Works and correctly computes diffusion-like behavior.
Stable Field Evolution: No runaway growth or numerical instability yet.
Doesn’t Yet Show Spontaneous Symmetry Breaking:
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Right now, the coefficients ϕ,π,Λ\phi, \pi, \Lambdaϕ,π,Λ are constant.
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To trigger SSB, we need to let them vary dynamically with Ψd\Psi_dΨd .
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Instead of fixed coefficients, let ϕd,πd\phi_d,\pi_dϕd,πd depend on local energy density.
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Implement: ϕd=ϕ0e−α∣Ψd∣2\phi_d = \phi_0 e^{-\alpha |\Psi_d|^2}ϕd=ϕ0e−α∣Ψd∣2 so that regions with high field intensity break symmetry differently.
Add Recursive Feedback Between Dimensions:
- Right now, we’re only solving for a single ddd-dimension.
- Next, we couple Ψd\Psi_dΨd to Ψd−1\Psi_{d-1}Ψd−1 and Ψd+1\Psi_{d+1}Ψd+1 .
and
We’ve now introduced dynamic feedback into the recursive evolution equation:
ϕd=ϕ0e−α∣Ψd∣2\phi_d = \phi_0 e^{-\alpha |\Psi_d|^2}ϕd=ϕ0e−α∣Ψd∣2
- High-field regions suppress ϕd\phi_dϕd , causing asymmetric field evolution.
- The field is no longer uniform, and localized structures are emerging.