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Scaling Laws: Microscopic and Macroscopic Behavior

  • Raffaele Esposito
Chapter

Abstract

The relation between microscopic and macroscopic descriptions of many-particle systems is discussed in terms of scaling laws, following the Boltzmann original ideas. Models where a complete mathematical treatment is possible are outlined.

Keywords

Transport Coefficient Hilbert Series Macroscopic Behavior Phase Point Macroscopic Description 
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References

  1. [1]
    Hilbert D., Mathematical Problems,Göttinger Nachroichten 1900, 253–297.Google Scholar
  2. [2]
    Boltzmann L., Über die Eigenshaften monzyklischer und anderer damit verwandter Systeme, in: “Wissenshaftliche Abhandlungen”, F.P. Hasenörl ed., vol. III, Chelsea, New York 1968.Google Scholar
  3. [3]
    Morrey C.B., On the derivation of the equations of hydrodynamics from statistical mechanics,Commun. Pure Applicata. Math. 8 (1955), 279–326.Google Scholar
  4. [4]
    Esposito R., Marra R., On the derivation of the incompressible Navier-Stokes equation for Hamiltonian particle systems, J. Stat. Phys. 74 (1993), 981–1004.CrossRefGoogle Scholar
  5. [5]
    Olla S., Varadhan S.R.S., Yau H.T., Hydrodynamical limit for a Hamiltonian system with weak noise, Commun. Math. Phys. 155 (1993), 523.CrossRefGoogle Scholar
  6. [6]
    Lanford O.E., Time Evolution of Large Classical systems,in: “Lecture Notes in Physics” 38 J. Moser ed., Springer, Berlin, Heidelberg, 1–111.Google Scholar
  7. [7]
    Hilbert D., Begründung der kinetischen Gastheorie,Mathematische Annalen 72 (1916/17), 331–407.Google Scholar
  8. [8]
    Caflisch R.E., The fluid dynamical limit of the nonlinear Boltzmann equation,Commun. Pure and Applicata. Math. 33 (1980), 651–666.Google Scholar
  9. [9]
    De Masi A., Esposito R., Lebowitz J.L., Incompressible Navier-Stokes and Euler Limits of the Boltzmann Equation, Commun. Pure and Applicata. Math. 42 (1989), 1189–1214.Google Scholar
  10. [10]
    Varadhan S.R.S., Nonlinear diffusion limit for a system with nearest neighbor interactions II, in: “Asymptotic Problems in Probability Theory: Stochastic Models and Diffusion on Fractals”, K.D. Elworthy and N. Ikeda eds., Pitman Research Notes in Math. 283, J. Wiley & Sons, New York 1994, 75–128.Google Scholar
  11. [11]
    Esposito R., Marra R., Yau H.T., Navier-Stokes equations for stochastic particle systems on the lattice, Commun. Math. Phys. 182 (1996), 395–456.CrossRefGoogle Scholar
  12. [12]
    Green M.S., Markoff random processes and the statistical mechanics of time-dependent phenomena. II. Irreversible processes in fluids,Jour. Chem. Phys. 22 (1954), 398–413.Google Scholar
  13. [13]
    Kubo R., Statistical-Mechanical theory of irreversible processes. I general theory and simple applications in magnetic and conduction problems,Jour. Phys. Soc. Jap. 12 (1957), 570–586.Google Scholar

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Raffaele Esposito
    • 1
    • 2
  1. 1.Dipartimento di Matematica Pura ed ApplicataUniversità degli Studi dell’AquilaItaly
  2. 2.Centro di Ricerche Linceo Interdisciplinare “Beniamino Segre”Italy

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