Materials Phonon Lifetime From Jcost

Phonon lifetime and its cost

A phonon is a quantized vibration of atoms in a crystal lattice. Its lifetime is the average time such a vibration persists before it scatters or decays, typically because of interactions with other phonons, defects, or electrons. In most materials at room temperature, acoustic phonon lifetimes fall in a range of about 1 to 100 picoseconds. One picosecond is one trillionth of a second, so these are extremely short-lived excitations.

The standard way to estimate a phonon lifetime involves the material's thermal conductivity, specific heat, and phonon group velocity, often through the Debye model. The Debye temperature, named after Peter Debye, is a material-specific temperature scale that characterizes its phonon spectrum. It enters equations for heat capacity and, in some models, for phonon lifetimes as well.

In Recognition Science, the framework models this quantity through a cost function, a measure of the price of recognition events in a discrete ledger of physical states. The framework proposes that the acoustic phonon lifetime scales as tau_ac = hbar / (J(phi) * k_B * T_Debye), where J is the framework's cost function, phi is the golden ratio, k_B is Boltzmann's constant, and T_Debye is the Debye temperature. Plugging in room temperature and a typical Debye temperature gives about 0.22 picoseconds, which lands in the correct order of magnitude.

What the module itself establishes is narrower and fully proved. It defines a domain cost as the cost function applied to the ratio of two real numbers, here interpreted as a mass divided by an energy. The module proves three general facts: this cost vanishes when the two numbers are equal, it is nonnegative for positive inputs, and the golden-ratio constant phi minus 3/2 is positive. These are universal properties of the cost function, not specific to phonons.

The module does not prove that the phonon lifetime formula is correct. Its own documentation states that the formula is a research note recording where the idea was meant to go, not a result. What would make it a theorem about phonons is a definition of the mass and energy in the material's own terms. The proved content is a template shared with many sibling modules, so the honest summary is: the framework proves general cost properties, and the phonon application remains a hypothesis with a clear falsifier.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

The phonon lifetime formula is not proved by the module; it is a research note. The cost function's general properties do not by themselves establish any material-specific result. No claim is made that the framework's phonon model matches experiment beyond an order-of-magnitude estimate.

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/Materials/Phonon_Lifetime_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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