Hydrodynamic limit of an exclusion process with vorticity

TL;DR

The paper constructs and analyzes a non-reversible exclusion process with Bernoulli product invariant measures in which the diffusive hydrodynamic limit features a non-symmetric diffusion matrix. It shows that the antisymmetric part affects the current but not the density evolution, and establishes a generalized gradient structure with an explicit discrete Hodge decomposition. In the presence of a weak external field, the mobility becomes symmetric and obeys an Einstein-type relation D(ρ) = σ(ρ) f''(ρ), with explicit solvable forms D(ρ) = I and σ(ρ) = ρ(1−ρ)I in two dimensions. The authors develop a robust hydrodynamic framework, including a replacement lemma and a functional-analytic current topology, to derive the scaling limits and discuss extensions to generalized gradient and weakly asymmetric models. Overall, the work provides a rigorous bridge between non-reversible lattice gases and macroscopic diffusive behavior, highlighting how antisymmetric dynamics shape current while preserving the density-driven hydrodynamics.

Abstract

We construct a non reversible exclusion process with Bernoulli product invariant measure and having, in the diffusive hydrodynamic scaling, a non symmetric diffusion matrix, that can be explicitly computed. The antisymmetric part does not affect the evolution of the density but it is relevant for the evolution of the current. Switching on a weak external field we obtain a symmetric mobility matrix that is related just to the symmetric part of the diffusion matrix by an Einstein relation. We argue that this fact is typical within a class of generalized gradient models. We consider for simplicity the model in dimension $d=2$, but a similar behavior can be also obtained in higher dimensions.

Figures (4)

• Figure 1: The possible configurations of particles on the two faces sharing the vertical edge $\{x,y\}$ when the lower vertex $x$ is occupied and the upper vertex $y$ is empty. Particles are denoted by $\bullet$, empty sites by $\circ$. Pairs of colored vertical square denote a configuration different from the one of the corresponding vertical edge in the case A. The two different colors are used to stress the fact that configurations associated to different colors may be different (case D).
• Figure 2: The graphical rules to determine the rates associated to a configuration of particles. Weights are associated to local configuration of particles as in the figure. Black arrows represent an unitary weight while blue arrows represent a weight $\alpha$. Weights concordantly oriented sum while weights oppositely oriented are subtracted.
• Figure 3: An example of a configuration with the weights associated according to the local rules of Figure \ref{['wei']}. The A inside a face means that the face is activated and weights $\alpha$ are associated to its edges counterclockwise.
• Figure 4: On the top we represent the value of \ref{['acea']} for two different configurations of particles (left and right respectively). Since \ref{['acea']} corresponds to the sum of 4 terms associated to directed edges around the face, we write near to each edge the corresponding contribution. If near an edge there is anything written, it means that the corresponding contribution is zero. At the bottom we have the same but for \ref{['acea2']}. Note that on the top, the edges form anticlockwise oriented cycles while, at the bottom, we have clockwise oriented cycles. The values of \ref{['acea']} and \ref{['acea2']} for a configuration of particles is obtained summing all the values associated to the edges on the corresponding figure.

Theorems & Definitions (26)

• Lemma 2.1
• proof
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Remark 3.1
• Remark 3.2
• Theorem 3.3
• proof