Astrophysics Picsimulation Lyapunov

The phi-ladder in plasma simulation

Particle-in-cell (PIC) simulation is a standard computational method in plasma physics. It tracks many individual charged particles through a grid of electromagnetic fields, letting researchers study phenomena like solar flares and fusion instabilities. A key diagnostic is the Lyapunov exponent, a number that measures how quickly nearby trajectories in the particle-field system diverge. A positive exponent signals chaos: tiny initial differences grow exponentially, limiting how far ahead a simulation can predict.

In the Recognition Science framework, this diagnostic takes a specific structural form. The framework models reality as a discrete ledger, a record of recognition events where each entry costs a forced amount. From that cost function, the framework derives the golden ratio phi, about 1.618, as the unique self-similar scaling. The module under discussion applies this scaling to PIC simulations: it defines a reference Lyapunov exponent at the coarsest resolution, then states that each finer resolution level, corresponding to doubling the number of macro-particles per Debye cell, divides the exponent by phi.

This is a definitional choice, not a measured law. The module defines lyapunovAt k as the reference exponent multiplied by phi raised to the power of negative k, where k is the resolution rung. It then proves three facts about this definition: the exponent is always positive, each step down the ladder multiplies the previous exponent by phi inverse, and the ratio between adjacent rungs equals phi inverse. These proofs are machine-checked, meaning the library of formal theorems verifies them with no gaps.

The empirical claim, from plasma simulation literature (Dawson 1983; Birdsall and Langdon 2004), is that adjacent doubling of particles per cell reduces numerical heating by phi squared. That ratio, phi squared, is the same canonical scaling the framework derives for the recognition lattice. The framework's contribution is not the measurement itself, but the structural prediction that the Lyapunov time, the inverse of the exponent, should sit on the phi-ladder across resolution levels.

What this establishes in plain language: within the framework, the chaotic sensitivity of a plasma simulation is not an arbitrary numerical artifact. It follows the same golden-ratio ladder that appears elsewhere in the framework, from Turing pattern thresholds to mass ratios. The module provides a formal certificate, a machine-checked guarantee that the defined exponents obey the ladder structure, which researchers could test against actual PIC runs.

/-- Reference Lyapunov exponent at `N_ppc` rung 0. -/
def referenceExponent : ℝ := 1

/-- 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_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

This module does not prove that real PIC simulations obey the phi-ladder; it defines a structural prediction and verifies the definition's internal consistency. The empirical heating reduction is cited from literature, not measured in the framework's library. The Lyapunov exponent here is a scalar diagnostic, not a full spectrum of exponents.

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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