Astrophysics Picsimulation Lyapunov Lyapunov At

A ladder of Lyapunov times

The Lyapunov exponent measures how quickly nearby trajectories in a dynamical system diverge. A positive exponent means chaos: small uncertainties grow exponentially fast, and its reciprocal, the Lyapunov time, is the horizon over which prediction fails. In particle-in-cell (PIC) plasma simulations, where thousands of macro-particles model a continuous plasma, this exponent depends on the numerical resolution, specifically the number of macro-particles per Debye cell, written N_ppc. The framework's declaration lyapunovAt defines a reference value for this exponent at a baseline resolution, then scales it down by a fixed factor at each higher resolution rung.

The scaling factor is the golden ratio φ, approximately 1.618. The definition sets the exponent at rung k to φ raised to the power −k, so each step to a higher rung multiplies the exponent by φ⁻¹, about 0.618. The library proves two properties of this definition: the exponent is always positive, and the ratio between adjacent rungs is exactly φ⁻¹. These are pure arithmetic facts about the definition, checked by the machine-checked library of formal theorems. They establish a clean geometric ladder: each resolution step shrinks the Lyapunov time by the same golden-ratio factor.

The empirical bench, attributed to Dawson 1983 and Birdsall-Langdon 2004, reports that doubling N_ppc reduces numerical heating by φ², which the framework identifies with the same φ² ratio appearing in its canonical scaling of the recognition lattice. This is a measured observation about PIC convergence, not a theorem. The framework's claim is structural: adjacent resolution levels sit on the φ-ladder, and the definition provides the formal certificate for that structure.

What lyapunovAt does not claim is more important. It does not assert that real PIC simulations actually produce Lyapunov exponents equal to this formula; the empirical match is a hypothesis with a named falsifier, namely a measurement of adjacent-resolution exponents whose ratio differs from φ⁻¹. It does not derive the Lyapunov exponent from first principles; the reference value is chosen as 1. And it does not claim that the golden ratio emerges from plasma physics itself, only that the framework's structural prediction places these simulation quantities on its ladder.

/-- Lyapunov exponent at PIC resolution rung `k` (higher rung = lower
numerical heating = smaller exponent). -/
def lyapunovAt (k : ℕ) : ℝ := referenceExponent * phi ^ (-(k : ℤ))

theorem lyapunovAt_pos (k : ℕ) : 0 < lyapunovAt k := by
  unfold lyapunovAt referenceExponent
  have : 0 < phi ^ (-(k : ℤ)) := zpow_pos Constants.phi_pos _
  linarith [this]

theorem lyapunovAt_adjacent_ratio (k : ℕ) :
    lyapunovAt (k + 1) / lyapunovAt k = phi⁻¹ := by
  rw [lyapunovAt_succ_ratio]
  field_simp [(lyapunovAt_pos k).ne']

structure PICLyapunovCert where
  lyapunov_pos : ∀ k, 0 < lyapunovAt k
  one_step_ratio : ∀ k, lyapunovAt (k + 1) = lyapunovAt k * phi⁻¹
  adjacent_ratio : ∀ k, lyapunovAt (k + 1) / lyapunovAt k = phi⁻¹

What this page does not claim

The formula does not predict the measured Lyapunov exponent of any real plasma simulation. The reference exponent value of 1 is a choice, not a derived quantity. The golden ratio is not claimed to be a property of plasma physics itself, only of the framework's structural prediction.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Astrophysics/PICSimulationLyapunov.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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