Materials Piezo Thermal Effect

The piezo thermal effect

The piezo thermal effect, known in physics as the pyroelectric effect, is the property of certain crystals to generate an electric voltage when heated or cooled. A classic example is tourmaline: warm it and it develops opposite charges on opposite faces. The effect has been studied since the 18th century, and it is the basis for infrared detectors and thermal imaging cameras. The size of the effect is measured by the pyroelectric coefficient, typically written p, which gives the change in electric polarization per unit change in temperature.

In the Recognition Science (RS) framework, this material property is modeled through the same cost function that governs recognition events. The framework defines a quantity called domainCost, a cost that measures the mismatch between two quantities m and e, here standing for material and energy. The cost is defined as J(m/e), where J is the forced cost function from the framework's central theorem. The Lean module for this topic proves three general facts about this cost: it is zero when m equals e, it is never negative when both inputs are positive, and a certain threshold value involving the golden ratio is positive.

What does this establish in plain language? The module proves that the cost function behaves sensibly: no negative costs, zero cost at perfect match, and a positive threshold. It does not prove anything specific to pyroelectric materials, because the module never defines what m and e mean in terms of a real crystal. The research note attached to the module suggests a structural approximation, p = J(phi) * epsilon_0 * (delta_P / delta_T), but this is a research idea, not a proved result.

The practical upshot is that RS provides a consistent formal language for thinking about material responses, but the bridge from the abstract cost to a measured pyroelectric coefficient remains open. For a reader, the module is a scaffold: it shows the shape a future derivation would take, but the material-specific definition of m and e is the missing piece.

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

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)

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

What this page does not claim

The module does not derive the pyroelectric coefficient from first principles. The research note's formula is not a proved result. No specific material's measured coefficient is reproduced by the framework.

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/Piezo_Thermal_Effect.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

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