A constitutive relation for diffusive transport stating that the particle flux J is proportional to the negative concentration gradient: J = −D ∇C, where D is the diffusion coefficient.

Principle

Matter flows from regions of higher concentration to lower concentration with a flux proportional to the local gradient; the proportionality constant D encodes medium and species properties.

Demonstration

In steady-state one-dimensional diffusion across a slab with constant D and concentration difference ΔC over thickness L, Fick's first law predicts constant flux J = −D (ΔC/L).

Misapplication

Applying the law to anomalous diffusion (subdiffusive or superdiffusive processes), strongly nonlocal transport, or at molecular scales where continuum assumptions and linear-response break down.

Consequence

Provides the constitutive input for Fick's second law (together with conservation) and yields boundary flux conditions used to solve diffusion PDEs in engineering and physics.

Reversal

Fick's second law describes the time evolution of concentration from the divergence of flux; reversing the constitutive relation would mean no gradient-driven flux and no diffusion-driven smoothing.

Boundary

Valid in the continuum limit for dilute species and near-linear response; excludes cases with concentration-dependent non-Fickian D without modification, and systems with strong advection or memory effects.

Semantic Tension

Tension exists between treating Fick's law as a local constitutive law versus modeling transport microscopically (random walks, continuous-time random walks) which may yield different macroscopic laws.

Synthesis

A linear constitutive law stating that diffusive flux is proportional to the negative local concentration gradient, valid for continuum, near-equilibrium diffusion of dilute species.