Fluctuation--response relations from an emergent $\mathbb{Z}_2$ symmetry in the rotating stochastic Landau model

What the math gives to ML

The paper derives fluctuation–response Ward identities from an emergent \(\mathbb{Z}_2\) symmetry of a coarse-grained stochastic action, without assuming detailed balance or thermal equilibrium. Its transferable asset is a practical distinction between response relations fixed by stochastic dynamics and the additional Einstein relation needed to determine the absolute noise scale. In neural-network training, this suggests an online noise-calibration controller that measures how parameter statistics respond to small perturbations and adjusts minibatch noise or injected Langevin noise toward a target fluctuation–response relation, while allowing nonequilibrium circulating dynamics rather than forcing reversible optimization.