Fick's laws of diffusion¶
Fick's laws relate diffusive flux to concentration gradient and concentration change to the divergence of that flux, yielding macroscopic transport equations from local conservation and a constitutive proportionality.
Core Idea¶
Fick's laws describe normal diffusion as transport down a concentration gradient and the resulting evolution of concentration in time. The first law states that diffusive flux J is proportional to the negative gradient of concentration c: J=−D∇c for an isotropic medium with diffusivity D. The minus sign encodes net movement from high toward low concentration. Combining this constitutive relation with local mass conservation yields the second law, ∂c/∂t=∇·(D∇c), which reduces to D∇²c when D is spatially constant.
Cross-Domain Echoes¶
See how this entry connects to another domain.
Scope of Application¶
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Molecular diffusion. Species spread through gases, liquids, and solids over scales where an effective continuum description is valid.
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Membranes. Steady and transient permeation is calculated with boundary concentrations and partitioning specified.
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Pharmacokinetics and tissue transport. Concentration profiles are modeled when reactions, flow, and heterogeneity are included as needed.
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Electrochemistry. Diffusion to electrodes is coupled with migration, reaction, and geometry under the chosen approximation.
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Materials engineering. Solute penetration, drying, and interdiffusion use concentration- and temperature-dependent coefficients where required.
Clarity¶
Fick's first law relates diffusive flux to a concentration gradient, while the second combines that relation with conservation to describe concentration evolution. Keeping constitutive law, conservation equation, initial conditions, and boundary conditions separate prevents a diffusion coefficient from being mistaken for the complete model. The standard form assumes normal diffusion and requires modification for advection, reactions, anisotropy, concentration-dependent diffusivity, or anomalous transport.
Manages Complexity¶
Fick's laws compress random molecular motion into concentration, gradient, diffusivity, flux, conservation, and boundary conditions. The first law supplies a local constitutive relation; the second propagates concentration through time. Steady, transient, constant-diffusivity, variable-diffusivity, anisotropic, reactive, and advective branches specify which form applies. The analyst can read direction of net flux, characteristic square-root time scale, and smoothing behavior without following individual particles.
Abstract Reasoning¶
Flux move. From a concentration gradient and diffusivity, infer the direction and magnitude of diffusive flux using the first law under its assumptions. Balance move. Combine flux divergence with conservation to derive time evolution of concentration through the second law. Boundary move. Solve with initial and boundary conditions appropriate to finite media, sources, sinks, or interfaces. Scaling move. Infer characteristic diffusion time from distance squared over diffusivity. Revision move. Replace the simple laws when diffusivity varies, transport is non-Fickian, or advection and reaction matter. Boundary move.
Knowledge Transfer¶
Within the home domain. Fick's laws transfer across gases, liquids, solids, membranes, physiology, electrochemistry, and materials when concentration gradients drive diffusive flux and conservation links flux divergence to concentration change. Diffusivity, gradient, boundary condition, geometry, source, and timescale retain physical roles. Beyond the home domain (B — shared abstract mechanism). Heat and momentum transport obey mathematically analogous gradient laws, sharing constitutive flux plus conservation. Molecular concentration and mass transport remain home-bound. Social “diffusion” is analogy unless a defensible field and flux exist, and advection, reactions, variable diffusivity, or anomalous transport can invalidate simple Fickian inference.
Relationships to Other Abstractions¶
Current abstraction Fick's laws of diffusion Domain-specific
Parents (1) — more general patterns this builds on
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Fick's laws of diffusion presupposes Diffusion Prime
Fick's laws of diffusion structurally presupposes Diffusion rather than being a subtype of it.
Hierarchy paths (3) — routes to 3 parentless roots
- Fick's laws of diffusion → Diffusion → Gradient
- Fick's laws of diffusion → Diffusion → Propagation
Neighborhood in Abstraction Space¶
Fick's laws of diffusion sits in a sparse region of the domain-specific corpus (69th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Boltzmann–Matano Analysis — 0.85
- Lattice Boltzmann Methods — 0.85
- Korteweg Stress — 0.84
- Reverse Diffusion — 0.84
- Kinematic Wave — 0.83
Computed from structural-signature embeddings · 2026-10-08