Skip to main content
SpringerLink
Log in

A Central Limit Theorem for Gibbsian Invariant Measures of 2D Euler Equations

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider canonical Gibbsian ensembles of Euler point vortices on the 2-dimensional torus or in a bounded domain of \(\mathbb {R}^2\). We prove that under the Central Limit scaling of vortices intensities, and provided that the system has zero global space average in the bounded domain case (neutrality condition), the ensemble converges to the so-called energy–enstrophy Gaussian random distributions. This can be interpreted as describing Gaussian fluctuations around the mean field limit of vortices ensembles of Caglioti et al. (Commun Math Phys 143(3):501–525, 1992) and Kiessling and Wang (J Stat Phys 148(5):896–932, 2012), and it generalises the result on fluctuations of Bodineau and Guionnet (Ann Inst H Poincaré Probab Stat 35(2):205–237, 1999). The main argument consists in proving convergence of partition functions of vortices.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume 55 of National Bureau of Standards Applied Mathematics Series. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. (1964)

  2. Albeverio, S., Ribeiro de Faria, M., Höegh-Krohn, R.: Stationary measures for the periodic Euler flow in two dimensions. J. Stat. Phys. 20(6), 585–595 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  3. Albeverio, S., Cruzeiro, A.B.: Global flows with invariant (Gibbs) measures for Euler and Navier–Stokes two-dimensional fluids. Commun. Math. Phys. 129(3), 431–444 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  4. Aubin, T.: Some Nonlinear Problems in Riemannian Geometry. Springer Monographs in Mathematics. Springer, Berlin (1998)

    Book  Google Scholar 

  5. Benfatto, G., Picco, P., Pulvirenti, M.: On the invariant measures for the two-dimensional Euler flow. J. Stat. Phys. 46(3–4), 729–742 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  6. Bessaih, H., Coghi, M., Flandoli, F.: Mean field limit of interacting filaments and vector valued non-linear PDEs. J. Stat. Phys. 166(5), 1276–1309 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  7. Bessaih, H., Coghi, M., Flandoli, F.: Mean field limit of interacting filaments for 3D Euler equations. J. Stat. Phys. 174(3), 562–578 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  8. Bodineau, T., Guionnet, A.: About the stationary states of vortex systems. Ann. Inst. H. Poincaré Probab. Stat. 35(2), 205–237 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  9. Caglioti, E., Lions, P.-L., Marchioro, C., Pulvirenti, M.: A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Commun. Math. Phys. 143(3), 501–525 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  10. Caglioti, E., Lions, P.-L., Marchioro, C., Pulvirenti, M.: A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. II. Commun. Mathun. Phys. 174(2), 229–260 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  11. Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions, volume 152 of Encyclopedia of Mathematics and its Applications, 2nd edn. Cambridge University Press, Cambridge (2014)

  12. Deutsch, C., Lavaud, M.: Equilibrium properties of a two-dimensional Coulomb gas. Phys. Rev. A 9, 2598–2616 (1974)

    Article  ADS  Google Scholar 

  13. Flandoli, F.: Weak vorticity formulation of 2D Euler equations with white noise initial condition. Commun. Part. Differ. Equ. 43(7), 1102–1149 (2018)

    Article  MathSciNet  Google Scholar 

  14. Flandoli, F., Luo, D.: \(\rho \)-white noise solution to 2D stochastic Euler equations. Probab. Theory Rel. Fields 175(3–4), 783–832 (2019)

    Article  MathSciNet  Google Scholar 

  15. Flandoli, F., Luo, D.: Point vortex approximation for 2D Navier–Stokes equations driven by space-time white noise (2019). arXiv e-prints arXiv:1902.09338

  16. Grotto, F.: Stationary Solutions of Damped Stochastic 2-dimensional Euler’s Equation (2019). arXiv e-prints arXiv:1901.06744

  17. Gunson, J., Panta, L.S.: Two-dimensional neutral Coulomb gas. Commun. Math. Phys. 52(3), 295–304 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  18. Janson, S.: Gaussian Hilbert Spaces. Cambridge Tracts in Mathematics, vol. 129. Cambridge University Press, Cambridge (1997)

  19. Kallenberg, O.: Foundations of Modern Probability. Probability and its Applications (New York). Springer, New York (2002)

    Google Scholar 

  20. Kiessling, M.K.-H.: Statistical mechanics of classical particles with logarithmic interactions. Commun. Pure Appl. Math. 46(1), 27–56 (1993)

    Article  MathSciNet  Google Scholar 

  21. Kiessling, M.K.-H., Wang, Y.: Onsager’s ensemble for point vortices with random circulations on the sphere. J. Stat. Phys. 148(5), 896–932 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  22. Kraichnan, R.H.: Remarks on turbulence theory. Adv. Math. 16, 305–331 (1975)

    Article  MathSciNet  Google Scholar 

  23. Leblé, T., Serfaty, S.: Fluctuations of two dimensional Coulomb gases. Geom. Funct. Anal. 28(2), 443–508 (2018)

    Article  MathSciNet  Google Scholar 

  24. Leblé, T., Serfaty, S., Zeitouni, O.: Large deviations for the two-dimensional two-component plasma. Commun. Math. Phys. 350(1), 301–360 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  25. Lions, P.-L.: On Euler equations and statistical physics. Cattedra Galileiana. [Galileo Chair]. Scuola Normale Superiore, Classe di Scienze, Pisa (1998)

  26. Marchioro, C., Pulvirenti, M.: Mathematical theory of incompressible nonviscous fluids. Applied Mathematical Sciences, vol. 96. Springer, New York (1994)

    Book  Google Scholar 

  27. Molčanov, S. A.: Diffusion processes, and Riemannian geometry. Uspehi Mat. Nauk, 30(1(181)):3–59, 1975. English translation: Russian Math. Surveys 30 (1975), no. 1, 1–63

  28. Neri, C.: Statistical mechanics of the \(N\)-point vortex system with random intensities on a bounded domain. Ann. Inst. H. Poincaré Anal. Non Linéaire 21(3), 381–399 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  29. Nowak, A., Sjögren, P., Szarek, T.Z.: Sharp estimates of the spherical heat kernel. J. Math. Pures Appl. 129(9), 23–33 (2019)

    Article  MathSciNet  Google Scholar 

  30. Onsager, L.: Statistical hydrodynamics. Nuovo Cimento (9), 6(Supplemento, 2 (Convegno Internazionale di Meccanica Statistica)):279–287 (1949)

  31. Polvani, L.M., Dritschel, D.G.: Wave and vortex dynamics on the surface of a sphere. J. Fluid Mech. 255, 35–64 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  32. Samuel, S.: Grand partition function in field theory with applications to Sine–Gordon field theory. Phys. Rev. D 18, 1916–1932 (1978)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126, Pisa, Italia

    Francesco Grotto

  2. Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italia

    Marco Romito

Authors
  1. Francesco Grotto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marco Romito
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marco Romito.

Additional information

Communicated by M. Hairer

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grotto, F., Romito, M. A Central Limit Theorem for Gibbsian Invariant Measures of 2D Euler Equations. Commun. Math. Phys. 376, 2197–2228 (2020). https://doi.org/10.1007/s00220-020-03724-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-020-03724-1

Search

Navigation