# Hydrodynamic Limit of Reversible Nongradient Systems

• Claude Kipnis
• Claudio Landim
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 320)

## Abstract

We investigate in this chapter the hydrodynamic behavior of reversible nongradient systems. To fix ideas we consider one of the simplest examples, the so called symmetric generalized exclusion process. This is the Markov process introduced in section 2.4 that describes the evolution of particles on a lattice with an exclusion rule that allows at most k particles per site. Here K is a fixed positive integer greater or equal than 2. The generator of this Markov process acts on cylinder functions as
$$\left( {{L_N}f} \right)\left( \eta \right) = \left( {1/2} \right)\sum\limits_{\mathop {x,y \in {T^d}}\limits_{|x - y| = 1} } {r\left( {\eta \left( x \right),\eta \left( y \right)} \right)} \left[ {f\left( {{\eta ^{x,y}}} \right) - f\left( \eta \right)} \right],$$
(0.1)
where r(a, b) = 1{a > 0, b < k} and η x, y is the configuration obtained from η moving a particle from x to y.

### Keywords

Entropy Covariance Lime Compressibility Univer

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1. Varadhan, S.R.S. (1994a): Nonlinear diffusion limit for a system with nearest neighbor interactions II. In K. D. Elworthy and N. Ikeda, editors, Asymptotic Problems in Probability Theory: Stochastic Models and Diffusion on Fractals. Volume 283 of Pitman Research Notes in Mathematics, pages 75–128. John Wiley and Sons, New YorkGoogle Scholar
2. Quastel, J. (1992): Diffusion of color in the simple exclusion process. Comm. Pure Appl. Math. XLV, 623–679Google Scholar
3. Yau, H.T. (1994): Metastability of Ginzburg—Landau model with a conservation law. J. Stat. Phys. 74, 705–742
4. Varadhan, S.R.S., Yau, H.T. (1997): Diffusive limit of lattice gases with mixing conditions, preprintGoogle Scholar
5. Yau, H.T. (1997): Logarithmic Sobolev inequality for generalized simple exclusion processes. Probab. Th. Rel. Fields 109, 507–538
6. Spohn, H., Yau, H.T. (1995): Bulk diffusivity of lattice gases close to criticality. J. Stat. Phys. 79, 231–241
7. Wick, W.D. (1989): Hydrodynamic limit of nongradient interacting particle processes. J. Stat. Phys. 89, 873–892
8. Landim, C., Yau, H.T. (1995): Large deviations of interacting particle systems in infinite volume. Comm. Pure Appl. Math. XLVIII, 339–379Google Scholar
9. Komoriya, K. (1997): Hydrodynamic limit for asymetric mean zero exclusion processes with speed change. preprintGoogle Scholar
10. Landim, C., Olla, S., Yau, H.T. (1997): First order correction for the hydrodynamic limit of asymmetric simple exclusion processes in dimension d 3, Comm. Pure Appl. Math. L, 149–203Google Scholar
11. Dobrushin, R.L. (1989): Caricatures of hydrodynamics. In B. Simon and A. Truman and I. M. Davies, editors, IXth International Congress on Mathematical Physics, pages 117132, Adam Hilger, BristolGoogle Scholar
12. Dobrushin, R.L., Pellegrinotti, A., Suhov, Yu.M., Triolo, L. (1988): One dimensional harmonic lattice caricature of hydrodynamics: second approximation. J. Stat. Phys. 52, 423–439
13. Dobrushin, R.L., Pellegrinotti, A., Suhov, Yu.M. (1990): One dimensional harmonic lattice caricature of hydrodynamics: a higher correction. J. Stat. Phys. 61, 387–402
14. Esposito, R., Marra, R. (1994): On the derivation of the incompressible Navier—Stokes equation for Hamiltonian particle systems. J. Stat. Phys. 74, 981–1004
15. Esposito, R., Marra, R., Yau, H.T. (1994): Diffusive limit of asymmetric simple exclusion. Rev. Math. Phys. 6, 1233–1267
16. Janvresse, E., Landim, C., Quastel, J., Yau, H.T. (1997): Relaxation to equilibrium of conservative dynamics I: zero range dynamics. PreprintGoogle Scholar
17. Landim, C., Yau, H.T. (1997): Fluctuation—dissipation equation of asymmetric simple exclusion processes. Probab. Th. Rel. Fields 108, 321–356
18. Yau, H.T. (1996): Logarithmic Sobolev inequality for lattice gases with mixing conditions. Commun. Math. Phys. 181, 367–408
19. Quastel, J., Yau, H.T. (1997): Lattice gases, large deviations and the incompressible NavierStokes equation, preprintGoogle Scholar
20. Landim, C., Olla, S., Volchan, S.B. (1997): Driven tracer particle and Einstein relation in one dimensional symmetric simple exclusion process. Resenhas IME—USP 3, 173–209
21. Janvresse, E. (1997): First order correction for the hydrodynamic limit of symmetric simple exclusion processes with speed change in dimension d 3, preprintGoogle Scholar
22. Landim, C., Olla, S., Yau, H.T. (1996): Some properties of the diffusion coefficient for asymmetric simple exclusion processes. Ann. Probab. 24, 1779–1807