Astrophysics Picsimulation Lyapunov Lyapunov At Succ Ratio

The resolution step rule

Particle-in-cell (PIC) simulations track how a plasma's particle and field quantities evolve in time. A Lyapunov exponent measures how quickly nearby starting states diverge: a larger exponent means faster divergence and a shorter predictability horizon. In the Recognition Science framework, the reference exponent is set to 1, and the exponent at resolution rung k is defined as φ^-k, where φ is the golden ratio, about 1.618.

The result lyapunovAt_succ_ratio proves that moving from one rung to the next multiplies the exponent by φ^-1. In symbols: lyapunovAt(k+1) = lyapunovAt(k) · φ^-1. This is a direct consequence of the definition and the algebraic properties of powers; the proof is machine-checked with no axioms beyond the standard logical core. A companion result states the same fact as a ratio: lyapunovAt(k+1) / lyapunovAt(k) = φ^-1.

What the declaration does not claim: it does not assert that real PIC simulations must obey this rule. The definition is a modeling choice, and the result is a fact about that definition. The empirical bench cited in the source (Dawson 1983; Birdsall-Langdon 2004) suggests that adjacent doubling of macro-particles reduces numerical heating by φ², which matches the φ² scaling of the recognition lattice, but that comparison is an empirical observation, not a proved consequence of the result.

The practical upshot for a reader: within the framework, the resolution ladder has a fixed geometric structure. If you know the exponent at one rung, you know it at every rung. The result gives a clean, checkable relation that any simulation claiming to follow the recognition lattice must satisfy.

theorem lyapunovAt_succ_ratio (k : ℕ) :
    lyapunovAt (k + 1) = lyapunovAt k * phi⁻¹ := by
  unfold lyapunovAt
  have hphi_ne : phi ≠ 0 := Constants.phi_ne_zero
  have : phi ^ (-((k : ℤ) + 1)) = phi ^ (-(k : ℤ)) * phi⁻¹ := by
    rw [show (-((k : ℤ) + 1)) = -(k : ℤ) + (-1 : ℤ) by ring]
    rw [zpow_add₀ hphi_ne]; simp
  have hcast : ((k + 1 : ℕ) : ℤ) = (k : ℤ) + 1 := by push_cast; ring
  rw [hcast, this]; ring

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

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

What this page does not claim

This result does not prove that real PIC simulations must follow the φ^-1 ratio. The empirical bench matching is an observation, not a result in the library.

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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