dC_dx ↔ dT_dx
Tier II — Similarity EMPIRICAL
- Object
- diffusing_medium
- Property
- ConcentrationGradient
- Context
- dilute
- Object
- heat_conducting_medium
- Property
- TemperatureGradient
- Context
- near_equilibrium
Sieve rationale
Structural equation-level identity (sign=+1): the canonical forms of EQ-FICK-DIFFUSION and EQ-FOURIER-HEAT are syntactically equal under rename, and dC_dx maps to dT_dx under the discovered bijection
Physical constraint filter
At least one variable is not a bond-graph port variable (role_a=gradient, role_b=gradient); Tellegen pairing is only defined for effort/flow.
Both equations are linear transport laws; for a single-variable coupling the Onsager matrix is 1x1 and symmetric by construction.
Energy conservation check is narrowly targeted at the Newton+Hooke → SHO composite in Medium m2; other composites require the Phase 7 Lyapunov/Noether machinery (not yet built).
Emergent properties
AI review board
2 REJECT vote(s); the coupling is demoted to conjectural per §12.2.
The coupling between Fick's law of diffusion and Fourier's law of heat conduction is a textbook example of structural analogy between linear transport equations: J = -D·(dC/dx) maps to q = -k·(dT/dx) under a consistent variable renaming. The concentration gradient and temperature gradient play identical structural roles (driving forces in their respective flux laws), and the tier-2 similarity classification with an identity transfer function accurately reflects this isomorphism rather than claiming the two quantities are literally identical. The Onsager reciprocity check correctly notes that a 1×1 coupling matrix is trivially symmetric, and the other constraint checks are properly marked not applicable. The Buckingham π groups are dimensionally consistent within each equation. No spurious emergent properties are claimed. This is a well-established and physically meaningful cross-domain analogy in irreversible thermodynamics.
Despite the structural similarity of Fick’s and Fourier’s laws, dC/dx (ConcentrationGradient) and dT/dx (TemperatureGradient) are different physical quantities with incompatible dimensions; an identity transfer function is therefore not physically plausible and is not supported by the semantic descriptors. A meaningful cross-domain link would require a named transform (e.g., nondimensionalization or phenomenological cross-coefficients such as Soret/Dufour relating appropriate forces/fluxes), not direct equality of the gradients. The physical-constraint filter’s Onsager “pass” concerns symmetry of transport matrices and does not validate variable identity, and the reported π-groups are trivial restatements of the base laws rather than genuine emergent properties.
The proposed coupling incorrectly equates a concentration gradient with a temperature gradient via an identity transfer function. While Fick's Law and Fourier's Law are structurally analogous and describe phenomena that are physically coupled in nature (e.g., thermodiffusion), a concentration gradient is not a temperature gradient. They are distinct physical quantities with different dimensions. This proposal confuses a valid mathematical analogy with an invalid physical identity, and as a result, no genuine emergent properties are discovered.