Gaussian-mixture kinetic neural solver / REPORT.md
Mechanism failed
Gaussian-mixture kinetic neural solver MVP
What was implemented
kinetic_mvp.py evaluates a positive 2D Gaussian mixture with softmax weights and softplus widths. It also contains a small homogeneous kinetic toy operator
L = gamma * (I - P)
where P projects onto the particle-number and quadratic-energy invariants. A response sweep adds the diagonal streaming term i(k p_x - omega) and compares mixture fitting against a coarse grid representation.
Run with:
/home/maxwelhelp/main/bin/python3 kinetic_mvp.py
The deterministic output is in results.json.
Numerical checks
| Prediction from mechanism | Predicted | Observed | Assessment |
|---|---:|---:|---|
| Gaussian mixture is positive | f(p) >= 0 | minimum 1.06e-20 | confirmed |
| Collision operator has number and energy null modes | 2 zero modes | 2 for every tested gamma | confirmed |
| Nonzero relaxation eigenvalues scale linearly with collision rate | median eigenvalue = gamma | 0.25, 0.50, 1.00, 2.00 for gamma 0.25, 0.5, 1, 2 | confirmed to machine precision |
| Ballistic/hydrodynamic transition at k v_th / gamma ~= 1 | transition near 1 | crude response FWHM for ratios 0.25, 0.5, 1, 2, 4 was 0.5, 4, 6, 6, 2; peak stayed at omega=0 | not confirmed |
The finite momentum box gives an integrated mixture mass of 0.8882, rather than one, because the reported quadrature box is only [-4,4]^2 and the mixture has non-negligible tail mass. The continuous mixture formula itself is normalized analytically; this is a truncation/quadrature diagnostic, not a positivity failure.
Representation comparison
On the same small synthetic mixture target, the fitted positive 3-component representation reached RMSE 1.35e-11, versus 2.19e-2 for the coarse 7x7 grid/nearest-cell representation. This supports the approximation advantage of the representation on this target, but it is not a neural-network training or full collision-integral benchmark.
Verdict
The conservation and relaxation-spectrum parts of the toy mechanism manifested clearly. The requested crossover prediction did not manifest with the chosen scalar response observable, and the implementation does not yet evaluate the full Bose gain-minus-loss collision integral with momentum/energy delta functions. Therefore this MVP does not establish the headline ballistic-to-hydrodynamic claim.