Heat in a crystal is carried by phonons — quantised lattice vibrations that travel at the speed of sound and randomize direction every time they scatter off another phonon, a defect or an isotope. That average travel distance between scatters is the phonon mean free path λ, typically tens to a few hundred nanometres in a real crystal near room temperature.
In a bulk crystal (thickness L ≫ λ) a phonon scatters thousands of times crossing the sample — a classic random walk that reproduces ordinary Fourier heat diffusion. Shrink the sample below λ (L ≲ λ, as in a nanowire or a thin film) and phonons increasingly fly straight from the hot face to the cold face with zero scattering events — ballistic transport — and Fourier's law itself breaks down.
Kn = λ / L (Knudsen number)
κ_eff / κ_bulk ≈ L / (L + λ) (Matthiessen's-rule size effect)
Kn ≪ 1 → diffusive, κ_eff ≈ κ_bulk
Kn ≫ 1 → ballistic, κ_eff collapses toward 0
- Sample thickness L — the nanowire/thin-film dimension heat must cross, hot face to cold face.
- Mean free path λ — set by material and temperature; longer at low temperature or in very pure/crystalline materials.
- Trails — orange straight segments are ballistic flights that reach the cold face with zero scattering; teal zig-zags are diffusive paths that scattered one or more times.
Real-world relevance: this size effect is why thermal conductivity measured on a nanowire or thin film is always lower than the bulk value of the same material, and why efficient thermoelectric materials are deliberately nanostructured to suppress phonon heat flow while leaving electron transport comparatively unhindered.