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

Every crystalline solid's thermal conductivity κ(T) traces the same shape: it rises steeply at low temperature, peaks, then slowly falls as temperature keeps climbing. This 2D simulator makes that curve emerge from a live ensemble of phonon particles random-walking inside a periodic grain: two independent, physically real Poisson scattering clocks — one set by the grain boundary, one by Umklapp phonon-phonon collisions whose rate grows steeply with temperature — compete for every phonon exactly the way Matthiessen's rule combines them analytically. Pick a real material preset (diamond, silicon or rock salt), drag temperature, grain-size, phonon-count and speed sliders, and watch the measured mean free path track the theoretical one while the κ(T) and Cv(T) charts trace out the conductivity peak in real time.