Hydrodynamic limit for some gradient and attractive spin models
Chiara Franceschini, Patrícia Gonçalves, Kohei Hayashi, Makiko Sasada
TL;DR
The paper studies the hydrodynamic limit for three gradient spin models with unbounded state spaces: a generalized KMP model, its discrete version, and a Harmonic model. By establishing a gradient structure and attractiveness, the authors prove that under diffusive time scaling, the empirical density converges to the solution of the heat equation with diffusion coefficient $D_ ext{gKMP}=D_ ext{dKMP}= frac{1}{2}$ and $D_ ext{Harm}= frac{1}{2 rak s}$, respectively. The proof combines the entropy method with a robust coupling framework, assuming initial measures are dominated by the invariant product measures. This work extends HDL results to unbounded-gradient systems, providing a foundation for future work on fluctuations and large deviations in non-equilibrium settings.
Abstract
We study the hydrodynamic limit for three gradient spin models: generalized Kipnis-Marchioro-Presutti (KMP), its discrete version and a family of harmonic models, under symmetric and nearest-neighbor interactions. These three models share some universal properties: occupation variables are unbounded, all these processes are of gradient type, their invariant measures are product with spatially homogeneous weights, and, notably, they are all attractive, meaning that the process preserves the partial order of measures along the dynamics. In view of hydrodynamics of large-scale interacting systems, dealing with processes taking values in unbounded configuration spaces is known to be a challenging problem. In the present paper, we show the hydrodynamic limit for all three models listed above in a comprehensive way, and show as a main result, that, under the diffusive time scaling, the hydrodynamic equation is given by the heat equation with model-dependent diffusion coefficient. Our novelty is showing the attractiveness for each model, which is crucial for the proof of hydrodynamics.