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

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Onsager's ergodic point vortex (sub-)ensemble is studied for N vortices which move on the 2-sphere 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 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.

Original languageEnglish (US)
Pages (from-to)896-932
Number of pages37
JournalJournal of Statistical Physics
Issue number5
StatePublished - Sep 2012

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


  • Continuum limit
  • Incompressible Euler fluid flow on S
  • Inequivalence of ensembles
  • Joyce-Montgomery mean-field theory
  • Miller-Robert theory
  • Onsager's ensemble
  • Point vortices
  • Random circulations
  • Turbulence


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