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October 2018 On the Green–Kubo formula and the gradient condition on currents
Makiko Sasada
Ann. Appl. Probab. 28(5): 2727-2739 (October 2018). DOI: 10.1214/17-AAP1369


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.


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Makiko Sasada. "On the Green–Kubo formula and the gradient condition on currents." Ann. Appl. Probab. 28 (5) 2727 - 2739, October 2018.


Received: 1 July 2017; Revised: 1 October 2017; Published: October 2018
First available in Project Euclid: 28 August 2018

zbMATH: 06974763
MathSciNet: MR3847971
Digital Object Identifier: 10.1214/17-AAP1369

Primary: 60K35
Secondary: 82C22

Keywords: diffusion coefficient , Gradient condition , Hydrodynamic limit , variational formula

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.28 • No. 5 • October 2018
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