## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 28, Number 5 (2018), 2727-2739.

### On the Green–Kubo formula and the gradient condition on currents

#### Abstract

In the diffusive hydrodynamic limit for a symmetric interacting particle system (such as the exclusion process, the zero range process, the stochastic Ginzburg–Landau model, the energy exchange model), a possibly nonlinear diffusion equation is derived as the hydrodynamic equation. The bulk diffusion coefficient of the limiting equation is given by the Green–Kubo formula and it can be characterized by a variational formula. In the case the system satisfies the gradient condition, the variational problem is explicitly solved and the diffusion coefficient is given from the Green–Kubo formula through a static average only. In other words, the contribution of the dynamical part of the Green–Kubo formula is $0$. In this paper, we consider the converse, namely if the contribution of the dynamical part of the Green–Kubo formula is $0$, does it imply the system satisfies the gradient condition or not. We show that if the equilibrium measure $\mu$ is product and $L^{2}$ space of its single site marginal is separable, then the converse also holds. The result gives a new physical interpretation of the gradient condition.

As an application of the result, we consider a class of stochastic models for energy transport studied by Gaspard and Gilbert in [*J. Stat. Mech. Theory Exp.* **2008** (2008) P11021; *J. Stat. Mech. Theory Exp.* **2009** (2009) P08020], where the exact problem is discussed for this specific model.

#### Article information

**Source**

Ann. Appl. Probab., Volume 28, Number 5 (2018), 2727-2739.

**Dates**

Received: July 2017

Revised: October 2017

First available in Project Euclid: 28 August 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1535443232

**Digital Object Identifier**

doi:10.1214/17-AAP1369

**Mathematical Reviews number (MathSciNet)**

MR3847971

**Zentralblatt MATH identifier**

06974763

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 82C22: Interacting particle systems [See also 60K35]

**Keywords**

Gradient condition variational formula diffusion coefficient hydrodynamic limit

#### Citation

Sasada, Makiko. On the Green–Kubo formula and the gradient condition on currents. Ann. Appl. Probab. 28 (2018), no. 5, 2727--2739. doi:10.1214/17-AAP1369. https://projecteuclid.org/euclid.aoap/1535443232