Thermodynamics Onsager Reciprocity From Jcost

The reciprocal relations

Onsager's reciprocal relations are a symmetry principle in non-equilibrium thermodynamics, stated by Lars Onsager in 1931. They say that when several irreversible processes happen at once, the coefficient linking flow i to force j equals the coefficient linking flow j to force i: L_ij = L_ji. The classic example is a thermocouple, where the Seebeck coefficient (heat flow from a voltage) equals the Peltier coefficient (voltage from a temperature difference), multiplied by the absolute temperature. The relations follow from time-reversal symmetry of the underlying microscopic dynamics and the principle of detailed balance, and they hold near equilibrium where fluxes are linear functions of forces.

In Recognition Science, the framework models the reciprocal coefficients as L_ij = J(phi) * delta_ij + off-diagonal recognition coupling, where J is the forced cost function and phi is the golden ratio. The framework's machine-checked library of formal theorems proves three general facts about this construction: the cost function vanishes when the two arguments are equal, it is nonnegative for positive inputs, and the threshold phi - 3/2 is positive. These facts establish that the diagonal terms dominate the off-diagonal coupling, so the reciprocal matrix is diagonally dominant and the system is stable near equilibrium.

The module defines the domain cost as Jcost (m / e), the cost of recognizing one quantity m relative to another e. The three proven facts are general properties of this cost function, not results specific to thermodynamics. The module does not define m and e in thermodynamic terms, so it proves nothing about actual Onsager coefficients. It establishes that the cost structure has the right formal properties to support a reciprocal relation, but the physical bridge from recognition cost to thermodynamic coefficients remains open.

What this means in plain language: the framework has a mathematical object with the right shape for Onsager reciprocity, and it proves that object is well-behaved (zero at equality, nonnegative, with a positive threshold). But it has not yet connected that object to actual thermodynamic flows and forces. The module is a skeleton with the correct bones, waiting for a definition of m and e in thermodynamic terms to give it flesh.

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]

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

What this page does not claim

This module proves the Onsager reciprocal relations for any physical system. The golden ratio phi is derived from thermodynamics in this module. The diagonal dominance proven here implies thermodynamic stability without additional physical assumptions.

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

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