Phonon Gas in a Grain — 2D Kinetic View of the Umklapp Peak

Material

Conditions

Temperature T 80 K

Grain size L (boundary path) 300 nm

Phonon count N 80

Animation speed 1.0×

Live Readouts

Conductivity κ(T)

— W/m·K

Theory ℓ

— nm

Measured ℓ (sim)

— nm

Heat capacity Cv/C_DP

—%

Boundary scatters/s

—

Umklapp scatters/s

—

κ(T) — dot = current T, dashed = peak T_peak

Debye C_v(T)/C_Dulong-Petit — dot = current T

How it works

This is a 2D kinetic-theory ensemble: each dot is a phonon moving in a straight line inside a periodic square domain whose physical width equals the grain size L. Two independent Poisson "scattering clocks" compete for every phonon, exactly reproducing Matthiessen's rule:

rate_total = 1/ℓ_boundary + 1/ℓ_Umklapp(T)
ℓ_Umklapp(T) ∝ exp(θ_D/3T) / T

Whichever clock fires first randomizes that phonon's direction — blue for a boundary hit, orange for an Umklapp phonon-phonon collision — and the "Measured ℓ (sim)" readout is the average distance actually travelled between scatter events in the running ensemble, which should track the analytic ℓ = 1/rate_total.

Heat is carried by this phonon gas with kinetic-theory conductivity:

κ = (1/3) · C_v(T) · v_sound · ℓ(T)

C_v(T) is the Debye heat capacity (rises as T³ at low T, saturates at the Dulong–Petit value 3nR). Only the fraction Cv(T)/Cv,max of phonons are drawn bright/active — the rest sit dim, representing modes not yet thermally populated. Drag the box to pan, scroll to zoom.

Active phonon Dormant mode Boundary scatter Umklapp scatter

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