Chemistry Glass Transition Relaxation Time

The relaxation time declaration

When a liquid is cooled fast enough to avoid crystallization, it does not freeze into a solid in the usual sense. Instead, its molecules become trapped in a disordered arrangement, and the material stiffens into a glass. The temperature at which this happens is called the glass transition temperature, Tg. A central question in glass science is how long it takes for the molecules in such a material to rearrange themselves, a quantity known as the relaxation time. Near Tg, this time can grow by many orders of magnitude, which is why a glass can flow imperceptibly over centuries.

The Recognition Science framework, a research program built around a machine-checked library of formal theorems, offers a specific model for this relaxation time. Its declaration, named relaxationTime, defines the relaxation time as a simple power law: τ = τ₀ × φ^n. Here, τ₀ is a base time scale, φ is the golden ratio (approximately 1.618), and n is an integer that depends on temperature relative to Tg. The framework's library proves that this quantity is always positive when the base time is positive, a basic but necessary consistency check. This is a model: a definitional choice that the framework makes, not a theorem derived from first principles.

The declaration does not claim that this power law is the measured behavior of any real glass. It does not predict a specific value for the base time τ₀, nor does it derive the exponent n from molecular structure. The framework's own documentation lists these as falsification criteria: if measured relaxation times do not follow the φ^n pattern, the model would be wrong. The declaration also does not claim to explain the famous fragility of glasses, the way viscosity shoots up differently for different materials. It merely establishes the mathematical form that the framework proposes for relaxation, leaving the empirical connection to be tested against experiment.

What the declaration does contribute is a concrete, testable target. By fixing the scaling law to a power of the golden ratio, the framework makes a sharp prediction that can be compared with data. The library proves the internal consistency of this choice, and the framework's own falsification criteria name exactly what would count as a failure. This combination of a precise model and an explicit list of what would refute it is the declaration's real content, and it is the standard by which the model should be judged.

/-- Relaxation time scaling with φ. -/
def relaxationTime (τ₀ : ℝ) (n : ℕ) : ℝ := τ₀ * Constants.phi ^ n

/-- Relaxation time is positive. -/
theorem relaxation_pos (τ₀ : ℝ) (hτ : 0 < τ₀) (n : ℕ) :
    0 < relaxationTime τ₀ n := by
  dsimp [relaxationTime]
  apply mul_pos hτ
  exact pow_pos Constants.phi_pos n

/-- Relaxation time scaling with φ. -/
def relaxationTime (τ₀ : ℝ) (n : ℕ) : ℝ := τ₀ * Constants.phi ^ n

What this page does not claim

The relaxation time model does not predict the measured relaxation time of any specific glass. The declaration does not derive the exponent n or the base time τ₀ from the framework's core principles. The model does not claim to explain the fragility index m or its variation across different glass formers.

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

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Derived articles

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