{
  "format_version": 1,
  "title": "Stationary density, saddle hitting, and escape in a quartic spectator diffusion",
  "model": {
    "potential": "U(x)=(x^2-1)^2/4",
    "drift": "-Uprime(x)",
    "noise_amplitude": "sqrt(2D)",
    "diffusion": "D=1/(4B)",
    "barrier_to_noise": "B=DeltaU/D; DeltaU=1/4",
    "time_variable": "s=lambda*v^2*t/(3H)",
    "field_variable": "x=phi/v",
    "physical_parameter_map": "D=3H^4/(8*pi^2*lambda*v^4)",
    "first_passage": {
      "initial": -1,
      "reflecting": -2,
      "absorbing": [
        0,
        0.5
      ]
    },
    "stationary": {
      "domain": [
        -2,
        2
      ],
      "boundaries": "both reflecting",
      "density": "exp(-U/D)/Z"
    },
    "scope": "Fixed-H additive Markov spectator model; no volume weighting, gravitational backreaction, or full-QFT equivalence claim."
  },
  "methods": {
    "continuum": "Nested composite trapezoidal quadrature of the exact backward-equation integral; time scaled by exp(-B).",
    "jump_rates": "q(j,j+/-1)=D/h^2*exp(-(U(j+/-1)-U(j))/(2D)); left reflecting outgoing rate doubled.",
    "independent_grid_solution": "Birth-death backward recurrence q_plus*d_next-q_minus*d_previous=-1, T(absorber)=0.",
    "simulation": "Uncensored continuous-time jump trajectories; exponential waiting time and rate-weighted next state. Exact for each fixed grid generator, then checked against the diffusion by grid refinement.",
    "random": "xoshiro128**, initial words [seed,0x9e3779b9,0x243f6a88,0xb7e15162], uint32 midpoint uniforms.",
    "uncertainty": "Sample-variance standard error and approximate normal 95 percent interval; neither is a rigorous finite-sample confidence bound. Grid bias reported separately."
  },
  "equilibrium": [
    {
      "barrier_to_noise": 2,
      "normalization_integral": 1.4109147025838793,
      "second_moment": 0.8521361507781037,
      "second_moment_refinement_change": 1.9517720772910252e-13
    },
    {
      "barrier_to_noise": 4,
      "normalization_integral": 0.947837832981343,
      "second_moment": 0.9176708607452315,
      "second_moment_refinement_change": 1.3322676295501878e-15
    }
  ],
  "asymptotic_boundary_check": [
    {
      "barrier_to_noise": 20,
      "results": [
        {
          "absorber": 0,
          "mean_time": 1099322196.686593,
          "ratio_to_kramers": 0.5100003667589131,
          "ratio_refinement_change": 5.551115123125783e-16
        },
        {
          "absorber": 0.5,
          "mean_time": 2198603878.4661913,
          "ratio_to_kramers": 1.0199819377384916,
          "ratio_refinement_change": 1.0694778396214133e-11
        }
      ]
    },
    {
      "barrier_to_noise": 40,
      "results": [
        {
          "absorber": 0,
          "mean_time": 527950434118532800,
          "ratio_to_kramers": 0.5048344745045726,
          "ratio_refinement_change": 9.880984919163893e-15
        },
        {
          "absorber": 0.5,
          "mean_time": 1055900866037045200,
          "ratio_to_kramers": 1.0096689469054514,
          "ratio_refinement_change": 2.7533531010703882e-14
        }
      ]
    },
    {
      "barrier_to_noise": 80,
      "results": [
        {
          "absorber": 0,
          "mean_time": 1.2366742335515734e+35,
          "ratio_to_kramers": 0.5023794769761414,
          "ratio_refinement_change": 6.661338147750939e-16
        },
        {
          "absorber": 0.5,
          "mean_time": 2.4733484671031103e+35,
          "ratio_to_kramers": 1.004758953952268,
          "ratio_refinement_change": 1.2656542480726785e-14
        }
      ]
    }
  ],
  "trajectory_comparison": [
    {
      "barrier_to_noise": 2,
      "absorber": 0,
      "continuum": {
        "mean_time": 18.42615844019114,
        "ratio_to_kramers": 0.5612818087211735
      },
      "relative_change_reflector_minus2_to_minus2p5": 4.832115652868836e-10,
      "refinements": [
        {
          "spacing": 0.1,
          "exact_grid_mean": 18.325018313106067,
          "relative_bias": -0.005488942657980578
        },
        {
          "spacing": 0.05,
          "exact_grid_mean": 18.400705877758185,
          "relative_bias": -0.0013813276660771324
        },
        {
          "spacing": 0.025,
          "exact_grid_mean": 18.419784814966484,
          "relative_bias": -0.00034590092369745577
        }
      ],
      "simulated_spacing": 0.05,
      "exact_simulated_grid_mean": 18.400705877758185,
      "sample": {
        "count": 6000,
        "seed": 20261107,
        "completed": 6000,
        "censored": 0,
        "total_jumps": 10976112,
        "mean_time": 18.373004932694663,
        "standard_error": 0.22924403683517494,
        "approximate_95_percent_interval": [
          17.92368662049772,
          18.822323244891606
        ],
        "mean_residual_in_standard_errors": -0.12083605508761282,
        "maximum_time": 181.06074166461588,
        "survival": [
          {
            "time": 9.200352938879092,
            "fraction": 0.6163333333333333,
            "binomial_standard_error": 0.006277825466878845
          },
          {
            "time": 18.400705877758185,
            "fraction": 0.3655,
            "binomial_standard_error": 0.006217043107458722
          },
          {
            "time": 36.80141175551637,
            "fraction": 0.12816666666666668,
            "binomial_standard_error": 0.004315475489874054
          }
        ]
      }
    },
    {
      "barrier_to_noise": 2,
      "absorber": 0.5,
      "continuum": {
        "mean_time": 35.02696200974322,
        "ratio_to_kramers": 1.066961225512645
      },
      "relative_change_reflector_minus2_to_minus2p5": 4.5046798945111204e-10,
      "refinements": [
        {
          "spacing": 0.1,
          "exact_grid_mean": 34.81566319998257,
          "relative_bias": -0.006032461784778061
        },
        {
          "spacing": 0.05,
          "exact_grid_mean": 34.97387367606035,
          "relative_bias": -0.0015156419694093085
        },
        {
          "spacing": 0.025,
          "exact_grid_mean": 35.013673414618324,
          "relative_bias": -0.0003793818921892682
        }
      ],
      "simulated_spacing": 0.05,
      "exact_simulated_grid_mean": 34.97387367606035,
      "sample": {
        "count": 6000,
        "seed": 20261112,
        "completed": 6000,
        "censored": 0,
        "total_jumps": 20688898,
        "mean_time": 34.61189166138956,
        "standard_error": 0.43346433626463854,
        "approximate_95_percent_interval": [
          33.762301562310874,
          35.46148176046825
        ],
        "mean_residual_in_standard_errors": -0.8350906508022138,
        "maximum_time": 298.03451298882067,
        "survival": [
          {
            "time": 17.486936838030175,
            "fraction": 0.6181666666666666,
            "binomial_standard_error": 0.00627211605559199
          },
          {
            "time": 34.97387367606035,
            "fraction": 0.36233333333333334,
            "binomial_standard_error": 0.00620547995577147
          },
          {
            "time": 69.9477473521207,
            "fraction": 0.129,
            "binomial_standard_error": 0.004327412621879268
          }
        ]
      }
    },
    {
      "barrier_to_noise": 4,
      "absorber": 0,
      "continuum": {
        "mean_time": 135.78297713738502,
        "ratio_to_kramers": 0.5597608604822858
      },
      "relative_change_reflector_minus2_to_minus2p5": 0,
      "refinements": [
        {
          "spacing": 0.1,
          "exact_grid_mean": 133.75243864371228,
          "relative_bias": -0.014954293509253671
        },
        {
          "spacing": 0.05,
          "exact_grid_mean": 135.27106301166626,
          "relative_bias": -0.00377009059980182
        },
        {
          "spacing": 0.025,
          "exact_grid_mean": 135.6547295440807,
          "relative_bias": -0.0009445042081714563
        }
      ],
      "simulated_spacing": 0.05,
      "exact_simulated_grid_mean": 135.27106301166626,
      "sample": {
        "count": 6000,
        "seed": 20261307,
        "completed": 6000,
        "censored": 0,
        "total_jumps": 40547108,
        "mean_time": 136.35596173765055,
        "standard_error": 1.7520179864404979,
        "approximate_95_percent_interval": [
          132.92200648422718,
          139.7899169910739
        ],
        "mean_residual_in_standard_errors": 0.6192280754996313,
        "maximum_time": 1494.3281943344168,
        "survival": [
          {
            "time": 67.63553150583313,
            "fraction": 0.613,
            "binomial_standard_error": 0.0062879646945573735
          },
          {
            "time": 135.27106301166626,
            "fraction": 0.3655,
            "binomial_standard_error": 0.006217043107458722
          },
          {
            "time": 270.5421260233325,
            "fraction": 0.136,
            "binomial_standard_error": 0.004425381339500586
          }
        ]
      }
    },
    {
      "barrier_to_noise": 4,
      "absorber": 0.5,
      "continuum": {
        "mean_time": 263.6999367219031,
        "ratio_to_kramers": 1.087094322134551
      },
      "relative_change_reflector_minus2_to_minus2p5": 0,
      "refinements": [
        {
          "spacing": 0.1,
          "exact_grid_mean": 259.69922763282773,
          "relative_bias": -0.015171445009842773
        },
        {
          "spacing": 0.05,
          "exact_grid_mean": 262.6919663605873,
          "relative_bias": -0.0038224141190402127
        },
        {
          "spacing": 0.025,
          "exact_grid_mean": 263.44745393339974,
          "relative_bias": -0.00095746245388608
        }
      ],
      "simulated_spacing": 0.05,
      "exact_simulated_grid_mean": 262.6919663605873,
      "sample": {
        "count": 6000,
        "seed": 20261312,
        "completed": 6000,
        "censored": 0,
        "total_jumps": 78121876,
        "mean_time": 262.69023699021113,
        "standard_error": 3.325904322196958,
        "approximate_95_percent_interval": [
          256.1714645187051,
          269.2090094617172
        ],
        "mean_residual_in_standard_errors": -0.0005199699716591624,
        "maximum_time": 2296.3214159908534,
        "survival": [
          {
            "time": 131.34598318029364,
            "fraction": 0.6126666666666667,
            "binomial_standard_error": 0.00628896152294137
          },
          {
            "time": 262.6919663605873,
            "fraction": 0.36616666666666664,
            "binomial_standard_error": 0.006219440474416876
          },
          {
            "time": 525.3839327211746,
            "fraction": 0.13516666666666666,
            "binomial_standard_error": 0.004413929445306998
          }
        ]
      }
    }
  ],
  "sources": [
    {
      "title": "Hofmann and Ivanyuk (2003), Eq. (2) and Fig. 3",
      "url": "https://arxiv.org/abs/nucl-th/0302022"
    },
    {
      "title": "Starobinsky and Yokoyama (1994), equilibrium Fokker-Planck density",
      "url": "https://doi.org/10.1103/PhysRevD.50.6357"
    },
    {
      "title": "Gillespie (1977), section IIIC, equations (21a)-(21b)",
      "url": "https://doi.org/10.1021/j100540a008"
    }
  ]
}