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Journal of Statistical Physics

, Volume 148, Issue 5, pp 896–932 | Cite as

Onsager’s Ensemble for Point Vortices with Random Circulations on the Sphere

  • Michael K.-H. KiesslingEmail author
  • Yu Wang
Article

Abstract

Onsager’s ergodic point vortex (sub-)ensemble is studied for N vortices which move on the 2-sphere \(\mathbb{S}^{2}\) with randomly assigned circulations, picked from an a-priori distribution. It is shown that the typical point vortex distributions obtained from the ensemble in the limit N→∞ are special solutions of the Euler equations of incompressible, inviscid fluid flow on \(\mathbb{S}^{2}\). These typical point vortex distributions satisfy nonlinear mean-field equations which have a remarkable resemblance to those obtained from the Miller-Robert theory. Conditions for their perfect agreement are stated. Also the non-random limit, when all vortices have circulation 1, is discussed in some detail, in which case the ergodic and holodic ensembles are shown to be inequivalent.

Keywords

Point vortices Random circulations Onsager’s ensemble Joyce-Montgomery mean-field theory Inequivalence of ensembles Continuum limit Incompressible Euler fluid flow on \(\mathbb{S}^{2}\) Turbulence Miller-Robert theory 

Notes

Acknowledgements

The authors gratefully acknowledge support from the NSF under grant DMS-0807705. Any opinions expressed in this paper are those of the authors and not necessarily those of the NSF. M.K. thanks David Dritschel for suggesting the investigation carried out in this paper, Edris Titi for a helpful correspondence on Euler’s equations of fluid flow on \(\mathbb{S}^{2}\), Greg Eyink and Marcello Lucia for helpful comments on the manuscript, and the latter for comments on [53] (see the Appendix). Lastly, we thank an anonymous referee for prompting us to expand the brief original Sects. 7.3.1 and 8 into the new 7.2.1, 7.3.1, 7.3.2, and 8.

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Authors and Affiliations

  1. 1.Department of MathematicsRutgers UniversityPiscatawayUSA

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