Astrophysics Picsimulation Lyapunov Lyapunov At Pos

The positivity theorem

A Lyapunov exponent measures how quickly nearby starting states in a dynamical system separate over time. A positive value signals chaos: tiny differences grow, making long-term prediction impossible. In particle-in-cell (PIC) simulations of plasma, researchers estimate this exponent for the coupled particle-field system, and the framework's library defines a sequence of such exponents, one for each resolution level of the simulation. The theorem lyapunovAt_pos proves that every value in this sequence is strictly greater than zero.

The sequence itself is simple. At the coarsest level, the reference exponent is set to 1. Each finer resolution level multiplies the previous exponent by the reciprocal of the golden ratio, approximately 0.618. So the exponents run 1, 0.618, 0.382, and so on, shrinking toward zero but never reaching it. The positivity theorem is the formal guarantee that this shrinking never crosses into zero or negative territory, for any natural-number resolution level. The proof is short: it relies on the fact that a positive number raised to any integer power stays positive.

The theorem also certifies that adjacent ratios are well-defined, since dividing by a positive number is always safe. That ratio, exactly the reciprocal of the golden ratio, is the structural prediction the framework attaches to PIC convergence: doubling the number of macro-particles per Debye cell reduces numerical heating by the square of that ratio. The positivity result is the foundation that makes the ratio statement meaningful.

What the theorem does not claim is equally important. It does not assert that real plasmas have positive Lyapunov exponents. It says nothing about whether PIC simulations accurately model any particular physical system. The theorem is about the defined sequence, not about the world. The empirical claim that PIC convergence follows this ratio is a separate, measured comparison, not part of the proof. The machine-checked result is a statement about arithmetic, cleanly bounded.

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_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']

What this page does not claim

The theorem does not claim that any real plasma has a positive Lyapunov exponent. The theorem does not claim that PIC simulations accurately model any particular physical system. The empirical PIC convergence comparison is a measured result, not a consequence of the proof.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: