Anomalous relaxation effects — where a system far from equilibrium reaches equilibrium faster than one closer to it — are well-demonstrated in small systems. The Mpemba effect (hot water freezing faster than cold) is the most famous example. Theoretical explanations typically describe relaxation through a single dominant exponential timescale. The far-from-equilibrium state happens to have less overlap with the slowest mode, so it relaxes faster.
In the thermodynamic limit, the single-exponential picture breaks (arXiv:2603.11326). For an antiferromagnetic Ising model on a square lattice, the discrete spectrum of relaxation timescales becomes continuous as the system grows. There is no single slowest mode to avoid. The relaxation is governed by a continuous distribution of timescales, and the anomalous effects — direct and inverse Mpemba, cooling-heating asymmetries, accelerated heating from precooling — must be understood through this continuous spectrum.
Near phase transitions, the slowest timescales can be characterized through the susceptibility associated with the order parameter of the metastable phase. This provides a handle: instead of tracking individual modes (impossible in the thermodynamic limit), track the susceptibility, which captures the aggregate behavior of the slow end of the spectrum.
The optimal protocols that produce the most pronounced anomalous relaxation can be predicted analytically from this framework and refined with Monte Carlo simulation. The anomalous effects are not fragile curiosities of small systems. They survive the thermodynamic limit — but the mechanism that explains them must change from mode avoidance to spectral shaping.