{
  "model": "Linearized isotropic gray Callaway operator in 3D angular space and one spatial Fourier mode. Single common speed, small perturbations; not graphite phonon spectrum, measured TTG optics or a fitted experiment.",
  "equation": "df/du=-i*mu*f-gamma_N*(f-P0*f-P1*f)-gamma_R*(f-P0*f); P0*f=<f>; P1*f=3*mu*<mu*f>",
  "angular_average": "integral from-1to1 dmu/2; Gauss-Legendre quadrature",
  "initial": "f(mu,0)=1 for the cosine Fourier component; no initial heat current",
  "rates": {
    "diffusive": {
      "gamma_N": 0,
      "gamma_R": 5
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    "ballistic": {
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      "gamma_R": 0
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    "weak_resistive": {
      "gamma_N": 0,
      "gamma_R": 0.05
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    "hydrodynamic": {
      "gamma_N": 20,
      "gamma_R": 0.05
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  "software": {
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    "scipy": "1.18.1",
    "matplotlib": "3.11.1"
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      "eigenvector_condition": 2.0152078968253186
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  },
  "ballistic_first_dip": {
    "u": 4.493409287285923,
    "amplitude": -0.2172336282112185
  },
  "checks": {
    "errors": {
      "diffusive_64_vs_128": 2.4980018054066022e-15,
      "ballistic_64_vs_128": 1.0972559666422299e-14,
      "weak_resistive_64_vs_128": 9.429956815409923e-15,
      "hydrodynamic_64_vs_128": 1.957323192414151e-12,
      "ballistic_vs_sinc": 6.938893903907228e-16,
      "zero_wavevector_energy": 1.687538997430238e-14
    },
    "hydrodynamic_limit": [
      {
        "normal_rate": 20,
        "error_to_telegraph": 0.0686314771135943
      },
      {
        "normal_rate": 80,
        "error_to_telegraph": 0.017896550873799044
      },
      {
        "normal_rate": 320,
        "error_to_telegraph": 0.0045217398236617745
      }
    ],
    "diffusion_limit": [
      {
        "resistive_rate": 5,
        "error_to_fourier": 0.012372101022694815
      },
      {
        "resistive_rate": 20,
        "error_to_fourier": 0.0008090966608581551
      }
    ],
    "conservation": "Both collision terms conserve angular average energy; normal term also conserves first angular moment; resistive term relaxes it at gamma_R.",
    "quadratic": "Angular-weight collision matrix negative semidefinite; streaming skew Hermitian. Quadratic norm nonincreasing, constant in ballistic case. This is a linearized model entropy proxy, not full measured sample entropy."
  }
}