Thermodynamics Boltzmann Hequation3

The H-function and its machine check

The Boltzmann H-theorem is a classical result in statistical mechanics. It concerns the H-function, a quantity that measures how far a gas is from equilibrium. The theorem states that H never increases: as molecules collide and exchange energy, H decreases monotonically until the gas reaches its equilibrium distribution. This is a statistical version of the second law of thermodynamics, and it is one of the few results that connects microscopic dynamics to macroscopic irreversibility.

In the framework of Recognition Science, the H-function is modeled using the cost function J. The framework's machine-checked library contains a module that defines the H-function as a sum over states: H = sum_k n_k * J(n_k/n_k^0), where n_k is the number of particles in state k and n_k^0 is the equilibrium number. The module proves three general facts about the cost function J: it vanishes when its two inputs are equal, it is nonnegative for positive inputs, and the golden-ratio-based threshold phi - 3/2 is positive. These are properties of J itself, not of the gas.

What the module does not do is prove the H-theorem for a physical gas. The module defines the cost as J(m/e) without specifying what m and e are in thermodynamic terms. The docstring is explicit: the paragraph about dH/dt is a research note, not a result. To make the module a theorem about Boltzmann's H-theorem, one would need to define m and e in the subject's own terms, such as particle number and equilibrium number. Until then, the module is a template, shared verbatim with 2383 sibling modules.

In Recognition Science, this is a deliberate division of labor. The framework proves the general properties of J once, in a central theorem, and then each subject module instantiates those properties with its own definitions. The Boltzmann module shows that the cost function has the right formal properties to serve as an H-function, but the physical bridge, connecting J to actual collision dynamics, remains open. This is a target for future work, not a completed derivation.

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 module does not prove the H-theorem for a physical gas; it proves properties of the cost function J alone. The module does not define what m and e are in thermodynamic terms. The module's content is not specific to this subject; it is shared verbatim with 2383 sibling modules.

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/Thermodynamics/BoltzmannHEquation3.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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