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 scaling limit for a stochastic PDE and the separation of phases
Download PDF
Download PDF
  • Published: June 1995

The scaling limit for a stochastic PDE and the separation of phases

  • T. Funaki1 

Probability Theory and Related Fields volume 102, pages 221–288 (1995)Cite this article

  • 442 Accesses

  • 63 Citations

  • Metrics details

Summary

We investigate the problem of singular perturbation for a reaction-diffusion equation with additive noise (or a stochastic partial differential equation of Ginzburg-Landau type) under the situation that the reaction term is determined by a potential with double-wells of equal depth. As the parameter ε (the temperature of the system) tends to 0, the solution converges to one of the two stable phases and consequently the phase separation is formed in the limit. We derive a stochastic differential equation which describes the random movement of the phase separation point. The proof consists of two main steps. We show that the solution stays near a manifoldM ε of minimal energy configurations based on a Lyapunov type argument. Then, the limit equation is identified by introducing a nice coordinate system in a neighborhood ofM ε.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Adams, R.A., Sobolev Spaces. Academic Press, New York London, 1975

    Google Scholar 

  2. Akhiezer, N.I., Glazman, I.M., Theory of Linear Operators in Hilbert Space, vol II. Pitman, London, 1981

    Google Scholar 

  3. Allen, S.M., Cahn, J.W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall.27 (1979), 1085–1095

    Google Scholar 

  4. Aronson, D.G., Weinberger, H.F., Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Goldstein J.A.: Partial Differential Equations and Related Topics (Lect. Notes Math., vol. 446, pp. 5–49) Berlin Heidelberg New York: Springer 1975

    Google Scholar 

  5. Benzi, R., Jona-Lasinio, G., Sutera, A., Stochastically perturbed Landau-Ginzburg equations. J. Stat. Phys.55 (1989), 505–522

    Google Scholar 

  6. Bertini, L., Presutti, E., Rüdiger, B., Saada, E., Dynamical fluctuations at the critical point: convergence to a non linear stochastic PDE. Theory Probab. Appl.38 (1993), 689–741

    Google Scholar 

  7. Bolthausen, E., Laplace approximations for sums of independent random vectors, Part II. Degenerate maxima and manifolds of maxima. Probab. Theory Relat. Fields76 (1987), 167–206

    Google Scholar 

  8. Brassesco, S., De Masi, A., Presutti, E., Brownian fluctuations of the instanton in the d=1 Ginzburg-Landau equation with noise, preprint, 1994

  9. Calderón, A.P.: Intermediate spaces and interpolation, the complex method. Studia Math.24 (1964), 113–190

    Google Scholar 

  10. Carr, J., Pego, R.L., Metastable patterns in solutions ofu t = ε2 u xx −f(u). Commun. Pure Appl. Math.XLII (1989), 523–576

    Google Scholar 

  11. Chiyonobu, T., On a precise Laplace-type asymptotic formula for sums of independent random vectors. In: Elworthy K.D., Ikeda N.: Asymptotic problems in probability theory, Wiener functionals and asymptotics (Pitman Research Notes in Math., vol. 284, pp. 122–135) Essex: Longman 1993

    Google Scholar 

  12. De Masi, A., Ferrari, P.A., Lebowitz, J.L., Reaction diffusion equations for interacting particle systems. J. Stat. Phys.44 (1986), 589–644

    Google Scholar 

  13. De Masi, A., Orlandi, E., Presutti, E., Triolo, L., Glauber evolution with Kac potentials. I. Mesoscopic and macroscopic limits, interface dynamics. Nonlinearity7 (1994), 633–696

    Google Scholar 

  14. De Masi, A., Orlandi, E., Presutti, E., Triolo, L., Motion by curvature by scaling nonlocal evolution equations. J. Stat. Phys.73 (1993), 543–570

    Google Scholar 

  15. De Masi, A., Pellegrinotti, A., Presutti, E., Vares, M.E., Spatial patterns when phases separate in an interacting particle system. Ann. Probab.22 (1994), 334–371

    Google Scholar 

  16. Diehl, H.W., Kroll, D.M., Wagner, H., The interface in a Ginsburg-Landau-Wilson model: Derivation of the drumhead model in the low-temperature limit. Z. PhysikB 36 (1980), 329–333

    Google Scholar 

  17. Eidel'man, S.D., Parabolic Systems (English translation). Noordhoff/North-Holland, Groningen/Amsterdam 1969

    Google Scholar 

  18. Evans, L.C., Soner, H.M., Souganidis, P.E., Phase transitions and generalized motion by mean curvature. Commun. Pure Appl. Math.XLV (1992), 1097–1123

    Google Scholar 

  19. Falconer, K.J., Differentiation of the limit mapping in a dynamical system. J. London Math. Soc.27 (1983), 356–372

    Google Scholar 

  20. Faris, W.G., Jona-Lasinio, G., Large fluctuations for a nonlinear heat equation with noise. J. Phys. A: Math. Gen.15 (1982), 3025–3055

    Google Scholar 

  21. Fife, P.C., McLeod, J.B., The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Rat. Mech. Anal.65 (1977), 335–361

    Google Scholar 

  22. Fife, P.C., Mathematical Aspects of Reacting and Diffusing Systems. Lect. Notes Biomath. vol. 28, Springer, Berlin Heidelberg New York, 1979

    Google Scholar 

  23. Fife, P.C., Dynamics of Internal Layers and Diffusive Interfaces. (CBMS-NSF Regional Conference Series in Applied Mathematics) Philadelphia: SIAM 1988

    Google Scholar 

  24. Fife, P.C., Hsiao, L., The generation and propagation of internal layers. Nonlinear Anal.12 (1988), 19–41

    Google Scholar 

  25. Flandoli, F., Deterministic attractors of ergodic stochstic flows. Scuola Normale Superiore Pisa, preprint 1990

  26. Freidlin, M., Functional integration and partial differential equations. Princeton Univ. Press, Princeton, 1985

    Google Scholar 

  27. Freidlin, M., Random perturbations of reaction-diffusion equations: The quasi-deterministic approximation. Trans. Amer. Math. Soc.305 (1988), 665–697

    Google Scholar 

  28. Freidlin, M., Semi-linear PDE's and limit theorems for large deviations. In: Hennequin P.L.: Ecole d'Eté de Probabilités de Saint-Flour XX-1990 (Lect. Notes Math., vol. 1527, pp. 2–109) Berlin Heidelberg New York: Springer 1992

    Google Scholar 

  29. Friedman, A., Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (New Jersey) 1964

    Google Scholar 

  30. Funaki, T., Random motion of strings and stochastic differential equations on the spaceC([0, 1],R d). In: Itô K.: Stochastic Analysis. Proc. Taniguchi Symp. 1982 (pp. 121–133) Amsterdam/Tokyo: North-Holland/Kinokuniya 1984

    Google Scholar 

  31. Funaki, T., The hydrodynamic limit for a system with interactions prescribed by Ginzburg-Landau type random Hamiltonian. Probab. Theory Relat. Fields90 (1991), 519–562

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  34. Funaki, T., Nagai, H., Degenerative convergence of diffusion process toward a submanifold by strong drift. Stochastics44 (1993), 1–25

    Google Scholar 

  35. Fusco, G., Hale, J.K., Slow-motion manifolds, dormant instability, and singular perturbations. J. Dynamics Diff. Eq.1 (1989), 75–94

    Google Scholar 

  36. Gärtner, J., Bistable reaction-diffusion equations and excitable media. Math. Nachr.112 (1983), 125–152

    Google Scholar 

  37. Gray, A., Tubes. Addison-Wesley, Reading (Massachusetts) 1990

  38. Hadeler, K.P., Rothe, F., Travelling fronts in nonlinear diffusion equations. J. Math. Biology2 (1975), 251–263

    Google Scholar 

  39. Hohenberg, P.C., Halperin, B.I., Theory of dynamic critical phenomena. Rev. Mod. Phys.49 (1977), 435–475

    Google Scholar 

  40. Iwata, K., An infinite dimensional stochastic differential equation with state spaceC(R). Probab. Theory Relat. Fields74 (1987), 141–159

    Google Scholar 

  41. Kato, T., Perturbation Theory for Linear Operators (second edition). Springer, Berlin Heidelberg New York 1976

    Google Scholar 

  42. Katzenberger, G.S., Solutions of a stochastic differential equation forced onto a manifold by a large drift, Ann. Probab.19 (1991), 1587–1628

    Google Scholar 

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

    Google Scholar 

  44. de Mottoni, P., Schatzman, M., Geometrical evolution of developped interfaces, preprint, 1989, to appear in Trans. Amer. Math. Soc.

  45. Mueller, C., Sowers, R., Random travelling waves for the KPP equation with noise, preprint, 1993

  46. Ohta, T., Jasnow, D., Kawasaki, K., Universal scaling in the motion of random interfaces. Phys. Rev. Lett.49 (1982), 1223–1226

    Google Scholar 

  47. Protter, M.H., Weinberger, H.F., Maximum Principles in Differential Equations, corrected reprint. Springer, Berlin Heidelberg New York 1984

    Google Scholar 

  48. Reed, M., Simon, B., Methods of Modern Mathematical Physics, I: Functional Analysis, revised and enlarged edition. Academic Press, New York London 1980

    Google Scholar 

  49. Spohn, H., Large Scale Dynamics of Interacting Particles. Springer, Berlin Heidelberg New York 1991

    Google Scholar 

  50. Spohn, H., Interface motion in models with stochastic dynamics. J. Stat. Phys.71 (1993), 1081–1132

    Google Scholar 

  51. Tanabe, H., Equations of Evolution. Pitman, London 1979

    Google Scholar 

  52. Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, Berlin Heidelberg New York 1988

    Google Scholar 

  53. Walsh, J.B., An introduction to stochastic partial differential equations. In: Hennequin P.L.: École d'Été de Probabilités de Saint-Flour XIV-1984 (Lect. Notes Math., vol. 1180, pp. 265–439) Berlin Heidelberg New York: Springer 1986

    Google Scholar 

  54. Yosida, K., Functional Analysis. Springer, Berlin Heidelberg New York 1965

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    T. Funaki

Authors
  1. T. Funaki
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research partially supported by Japan Society for the Promotion of Science

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Funaki, T. The scaling limit for a stochastic PDE and the separation of phases. Probab. Th. Rel. Fields 102, 221–288 (1995). https://doi.org/10.1007/BF01213390

Download citation

  • Received: 26 September 1994

  • Accepted: 10 January 1995

  • Issue Date: June 1995

  • DOI: https://doi.org/10.1007/BF01213390

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

Mathematics Subject Classification

  • 60H15
  • 60K35
  • 35R60
  • 82C24
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