Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
The hydrodynamic limit for a system with interactions prescribed by Ginzburg-Landau type random Hamiltonian
Download PDF
Download PDF
  • Published: December 1991

The hydrodynamic limit for a system with interactions prescribed by Ginzburg-Landau type random Hamiltonian

  • Tadahisa Funaki1 

Probability Theory and Related Fields volume 90, pages 519–562 (1991)Cite this article

  • 92 Accesses

  • 6 Citations

  • Metrics details

Summary

As a microscopic model we consider a system of interacting continuum like spin field overR d. Its evolution law is determined by the Ginzburg-Landau type random Hamiltonian and the total spin of the system is preserved by this evolution. We show that the spin field converges, under the hydrodynamic space-time scalling, to a deterministic limit which is a solution of a certain nonlinear diffusion equation. This equation describes the time evolution of the macroscopic field. The hydrodynamic scaling has an effect of the homogenization on the system at the same time.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Albeverio, S., Röckner, M.: Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms. Probab. Theory Relat. Fields89, 347–386 (1991)

    Google Scholar 

  2. De Masi, A., Ianiro, N., Pellegrinotti, A., Presutti, E.: A survey of the hydrodynamical behavior of many-particle systems. In: Lebowitz, J.L., Montroll, E.W. (eds.) Nonequilibrium phenomena II. From stochastics to hydrodynamics, pp. 123–294. Amsterdam: North-Holland 1984

    Google Scholar 

  3. Fritz, J.: On the hydrodynamic limit of a Ginzburg-Landau lattice model. The law of large numbers in arbitrary dimensions. Probab. Theory Relat. Fields81, 291–318 (1989)

    Google Scholar 

  4. Fritz, J.: Hydrodynamics in a symmetric random medium. Commun. Math. Phys.125, 13–25 (1989)

    Google Scholar 

  5. Fritz, J.: On the diffusive nature of ontropy flow in infinite systems: Remarks to a paper by Guo-Papanicolaou-Varadhan. Commun. Math. Phys.133, 331–352 (1990)

    Google Scholar 

  6. Fritz, J., Maes, C.: Derivation of a hydrodynamic equation for Ginzburg-Landau models in an external field. J. Stat. Phys.53, 1179–1206 (1988)

    Google Scholar 

  7. Funaki, T.: Derivation of the hydrodynamical equation for one-dimensional GinzburgLandau model. Probab. Theory Relat. Fields82, 39–93 (1989)

    Google Scholar 

  8. Funaki, T.: Regularity properties for stochastic partial differential equations of parabolic type. Osaka J. Math.28, 495–516 (1991)

    Google Scholar 

  9. Funaki, T.: The reversible measures of multi-dimensional Ginzburg-Landau type continuum model. Osaka J. Math.28, 463–494 (1991)

    Google Scholar 

  10. Guo, M.Z., Papanicolaou, G.C., Varadhan, S.R.S.: Nonlinear diffusion limit for a system with nearest neighbor interactions. Commun. Math. Phys.118, 31–59 (1988)

    Google Scholar 

  11. Hohenberg, P.C., Halperin, B.I.: Theory of dynamic critical phenomena. Rev. Mod. Phys.49, 435–479 (1977)

    Google Scholar 

  12. Jona-Lasinio, G., Mitter, P.K.: On the stochastic quantization of field theory. Commun. Math. Phys.101, 409–436 (1985)

    Google Scholar 

  13. Kawasaki, K.: Kinetics of Ising models. In: Domb, C., Green, M.S. (eds.) Phase transitions and critical phenomena, vol. 2, pp. 443–501. London New York: Academic Press 1972

    Google Scholar 

  14. Kawasaki, K., Ohta, T.: Kinetic drumhead model of interface, I. Prog. Theor. Phys.67, 147–163 (1982)

    Google Scholar 

  15. Künseh, H.: Decay of correlations under Dobrushin's uniqueness condition and its applications. Commun. Math. Phys.84, 207–222 (1982)

    Google Scholar 

  16. Lions, J.L., Magenes, E.: Non-homogeneous boundary value problems and applications, vol. I. Berlin Heidelberg New York: Springer 1972

    Google Scholar 

  17. Rezakhanlou, F.: Hydrodynamic limit for a system with finite range interactions. Commun. Math. Phys.129, 445–480 (1990)

    Google Scholar 

  18. Spohn, H.: Large scale dynamics of interacting particles, Part B: Stochastic lattice gases. (preprint, 1989)

  19. Stroock, D.W., Varadhan, S.R.S.: Multidimensional diffusion processes. Berlin Heidelberg New York: Springer 1979

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, School of Science, Nagoya University, 464-01, Nagoya, Japan

    Tadahisa Funaki

Authors
  1. Tadahisa Funaki
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Funaki, T. The hydrodynamic limit for a system with interactions prescribed by Ginzburg-Landau type random Hamiltonian. Probab. Th. Rel. Fields 90, 519–562 (1991). https://doi.org/10.1007/BF01192142

Download citation

  • Received: 12 December 1990

  • Revised: 19 August 1991

  • Issue Date: December 1991

  • DOI: https://doi.org/10.1007/BF01192142

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Time Evolution
  • Stochastic Process
  • Probability Theory
  • Diffusion Equation
  • Mathematical Biology
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature