Teaching non-equilibrium relaxation, phase change, and statistical reasoning through the Mpemba effect: a graded sequence of computational models

The Mpemba effect, the observation that hot water sometimes freezes before an identical cold sample, is a famous puzzle that students find immediately engaging. We use it as the organizing thread for a graded sequence of computational models, designed as classroom material, that brings together standard but rarely combined topics: Markovian relaxation and eigenmode decomposition, moving-boundary (Stefan) phase change, and the elementary statistics of effect size and statistical power. Built in a single accessible framework, the sequence clarifies why an effect that is rigorous in the abstract is so elusive in a bucket of water. In a Markovian master-equation model the effect appears as a crossing of two relaxation trajectories. The crossing is genuine but requires a specific geometric relation between the initial state and the slowest relaxation mode, so that with thermal (Boltzmann) initial conditions it is absent in a three-state system and confined to a thin parameter region in a four-state system. When a one-dimensional Stefan freezing front with stochastic nucleation is added, the random spread of freezing times (σ/μ ≈ 7%) far exceeds the mean hot–cold difference. The resulting effect size, d ≈ 0.03, would require thousands of paired trials to resolve, which restates quantitatively the recently argued stochastic origin of the effect. Adding convection and evaporative mass loss reproduces the classic non-monotonic (inverted-U) dependence of freezing time on initial temperature reported by Mpemba and Osborne, but only within a narrow, parameter-sensitive window. Rather than advancing new physics, the sequence gives students a coherent, hands-on route from an abstract master equation to a kitchen-scale freezing model, and it uses a quantitative signal-to-noise argument to show why a well-understood relaxation phenomenon is so hard to demonstrate in practice. Learning outcomes, suggested exercises, and all code are provided.