Hydrodynamic limit for some gradient and attractive spin models

TL;DR

This paper establishes the hydrodynamic limit for three gradient spin models with unbounded variables, demonstrating that their large-scale behavior is governed by the heat equation with specific diffusion coefficients, emphasizing the role of attractiveness.

Contribution

The paper proves the hydrodynamic limit for three models, highlighting their attractiveness and unbounded state space, which was previously challenging in such systems.

Findings

Hydrodynamic limit shown for all three models.

Large-scale behavior governed by heat equation.

Attractiveness proved for each model.

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.