Gradient flow of the infinite-volume free energy for lattice systems of continuous spins
Ronan Herry, Thomas Leblé
TL;DR
This work develops a rigorous gradient-flow framework for the infinite-volume free energy of lattice spin systems with spins in a compact manifold, connecting the gradient flow to the infinite-volume Langevin diffusion and to a hierarchy of infinite-volume Fokker–Planck–Kolmogorov equations. The authors construct the gradient flow via an infinite-volume JKO scheme with a stationarization step, and they build the infinite-volume diffusion as the limit of finite-volume diffusions. They prove that gradient-flow trajectories and diffusion laws solve the same infinite-volume FPK equations in dual/weak form, and they establish an Evolution Variational Inequality that yields uniqueness and a contraction property. Under positive curvature and small inverse temperature, the free energy decays and the dynamics converge exponentially to the unique minimizer in both free energy and a specific Wasserstein distance, highlighting a robust long-time behavior. The results extend gradient-flow and optimal-transport techniques to infinite-volume interacting spin systems, offering a unified variational-diffusive description and providing regularity and uniqueness results beyond previous finite-volume settings.
Abstract
We consider an infinite lattice system of interacting spins living on a smooth compact manifold, with short- but not necessarily finite-range pairwise interactions. We construct the gradient flow of the infinite-volume free energy on the space of translation-invariant spin measures, using an adaptation of the variational approach in Wasserstein space pioneered by Jordan, Kinderlehrer, and Otto. We also construct the infinite-volume diffusion corresponding to the so-called overdamped Langevin dynamics of the spins under the effect of the interactions and of thermal agitation. We show that the trajectories of the gradient flow and of the law of the spins under this diffusion both satisfy, in a weak sense, the same hierarchy of coupled parabolic PDE's, which we interpret as an infinite-volume Fokker-Planck-Kolmogorov equation. We prove regularity of weak solutions and derive an Evolution Variational Inequality for regular solutions, which implies uniqueness. Thus, in particular, the trajectories of the gradient flow coincide with those obtained from the Langevin dynamics. Concerning the long-time evolution, we check that the free energy is always non-increasing along the flow and that moreover, if the Ricci curvature of the spin space is uniformly positive, then at high enough temperature the dynamics converges exponentially, in free energy and in specific Wasserstein distance, to the unique minimizer of the infinite-volume free energy.