Journal of Statistical Physics

, Volume 74, Issue 5–6, pp 981–1004 | Cite as

On the derivation of the incompressible Mavier-Stokes equation for Hamiltonian particle systems

  • R. Esposito
  • R. Marra


We consider a Hamiltonian paticle system interacting by means of a pair potetial. We look at the behavior of the system on a space scale of order ε-1, times of order ε-2 and mean velocities of order ε, with ε a scale parameter. Assuming that the phase space density of the particles is give by a series in ε (the analog of the Chapman-Enskog expansion), the behavior of the system under this rescaling is described, to the lowest order in ε, by the incompressible Navier-Stokes equations. The viscosity is given in terms of microscopic correlations, and its expression agrees with the Green-Kubo formula.

Key Words

Hydrodynamic limit incompressible Navier-Stokes equations particle systems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    C. B. Morrey, On the derivation of the equations of hydrodynamics from statistical mechanics,Commun. Pure Appl. Math. 8:279–290 (1955).Google Scholar
  2. 2.
    A. De Masi, N. Ianiro, A. Pelegrinotti, and E. Presutti, A survey of the hydrodynamical behavior of many particle system, inStudies in Statistical Mechanics XI, E. W. Montroll and J. Lebowitz, eds. (North-Holland, Amsterdam, 1984), pp. 123–294.Google Scholar
  3. 3.
    S. Olla, S. R. S. Varadhan, and H. T. Yau, Hydrodynamical liomit for a Hamiltonian system with weak noise.Commun. Math. Phys. 155:523–560 (1993).Google Scholar
  4. 4.
    H. Spohn,Large Scalle Dynamics of Interacting Particles (Springer-Verlag, New York, 1991.Google Scholar
  5. 5.
    R. E. Caflisch, The fluid dynamic limit of the nonlilnear Boltzmann equation,Commun. Pure Appl. Math. 33:651–666 (1980).Google Scholar
  6. 6.
    C. Cercignani,The Boltzmann Equation and Its Applications, (Springer-Verlag, New York, 1988).Google Scholar
  7. 7.
    A. De Masi, R. Esposito and J. L. Lebowitz, Incompressible Navier-Stokes and Euler limits of the Boltzmann equation,Commun. Pure Appl. Math. 42:1189–1214 (1989).Google Scholar
  8. 8.
    C. Bardos, F. Golse, and D. Levermore, Fluid dynamical limits of kinetic equaitons I. Formal derivations.J. Stat. Phys. 63:323–344 (1991).Google Scholar
  9. 9.
    H. S. Green,J. Chem. Phys. 22:398 (1954).Google Scholar
  10. 10.
    R. Kubo,J. Phys. Soc. Jpn. 12:570 (1957).Google Scholar
  11. 11.
    H. S. Green,Theories of Transsport in Fluids, J. Math. Phys. 2:344–348 (1961).Google Scholar
  12. 12.
    D. N. Zubarev,Nonequilibrium Statistical Thermodynamics (Consultants Bureau, New York, 1974).Google Scholar
  13. 13.
    G. Eyink, unpublished notes.Google Scholar
  14. 14.
    R. Esposito, R. Marra, and H. T. Yau, Diffusive limit of the asymmetric simple exclusion,Rev. Math. Phys., to appear.Google Scholar
  15. 15.
    A. De Maasi and E. Presutti,Mathematical Methods for Hydrodynamic Limits (Springer-Verlag, Berlin, 1991).Google Scholar
  16. 16.
    H. Rost, Non-equilibrium behavior of a many-particle system: Density-profile and local equilibrium,Z. Wahrsch. Verw. Geb. 58:41–54 (1981); F. Rezakanlou, Hydrodynamic limit for attactive particle systems on ℤd,Commun. Math. Phys. 140:417–448 (1991).Google Scholar
  17. 17.
    S. R. S. Varadhan, Nonlinear diffusion limit for a system with nearest neighbor interaction II, inProceedings Taniguchi Symposium, Kyoto (1990).Google Scholar
  18. 18.
    H. T. Yau, Relative entropy and hydrodynamics of Ginsburg-Landau models.Lett. Math. Phys 22:63–80 (1991).Google Scholar
  19. 19.
    S. Keatz, J. L. Lebowitz, and H. Spohn, Nonequilibrium steady state of stochastic lattice gas model of fast ionic conductors,J. Stat. Phys. 34:497–538 (1984).Google Scholar
  20. 20.
    H. Spohn, Equilibrium fluctuations for interacting Brownian particles,Commun. Math. Phys. 103:1–33 (1986).Google Scholar
  21. 21.
    E. Presutti, A mechanical definition of the thermodynamic pressure,J. Stat. Phys. 13:301 (1975).Google Scholar

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • R. Esposito
    • 1
  • R. Marra
    • 2
  1. 1.Dipartimento di MatematicaUniversità di Roma Tor VergataRomeItaly
  2. 2.Dipartimento di FisicaUniversità di Roma tor VergataRomeItaly

Personalised recommendations