The Annals of Applied Probability

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

Makiko Sasada

Full-text: Open access


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C22: Interacting particle systems [See also 60K35]

Gradient condition variational formula diffusion coefficient hydrodynamic limit


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.

Export citation


  • [1] Gaspard, P. and Gilbert, T. (2008). On the derivation of Fourier’s law in stochastic energy exchange systems. J. Stat. Mech. Theory Exp. 2008 P11021.
  • [2] Gaspard, P. and Gilbert, T. (2009). Heat transport in stochastic energy exchange models of locally confined hard spheres. J. Stat. Mech. Theory Exp. 2009 P08020.
  • [3] Gaspard, P. and Gilbert, T. (2017). Dynamical contribution to the heat conductivity in stochastic energy exchanges of locally confined gases. J. Stat. Mech. Theory Exp. 2017 043210.
  • [4] Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 320. Springer, Berlin.
  • [5] Lu, S. (1994). Equilibrium fluctuations of a one-dimensional nongradient Ginzburg–Landau model. Ann. Probab. 22 1252–1272.
  • [6] Nagahata, Y. (1998). The gradient condition for one-dimensional symmetric exclusion processes. J. Stat. Phys. 91 587–602.
  • [7] Sasada, M. (2016). Thermal conductivity for stochastic energy exchange models. Available at arXiv:1611.08866.
  • [8] Spohn, H. (1991). Large Scale Dynamics of Interacting Particles. Springer, Berlin.
  • [9] Varadhan, S. R. S. (1993). Nonlinear diffusion limit for a system with nearest neighbor interactions. II. In Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals (Sanda/Kyoto, 1990). Pitman Res. Notes Math. Ser. 283 75–128. Longman Sci. Tech., Harlow.