Cooling rate and glassy behavior in the Fermi--Pasta--Ulam system

What the math gives to ML

The paper identifies a dynamical freeze-out mechanism during cooling: below a weak stochastic threshold, the FPU energy no longer tracks the bath temperature and approaches a nonzero residual energy. Its central quantitative result is the scaling E0 approximately proportional to (xi N)^(2/3), where xi is the cooling rate and N is the number of dynamical degrees of freedom. This can transfer to stochastic neural-network optimization as a freeze-out-aware noise scheduler: cool gradient noise slowly enough that residual parameter fluctuations remain below a prescribed tolerance. The key falsifiable signature is a 2/3 power law for residual optimizer variance and a collapse across model sizes when plotted against xi times effective dimension.