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Equilibrium energy spectrum of point vortex motion with remarks on ensemble choice and ergodicity

J. G. Esler*

  • Department of Mathematics, University College London, London WC1E 6BT, United Kingdom

  • *j.g.esler@ucl.ac.uk
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Phys. Rev. Fluids 2, 014703 – Published 11 January, 2017

DOI: https://doi.org/10.1103/PhysRevFluids.2.014703

Abstract

The dynamics and statistical mechanics of N chaotically evolving point vortices in the doubly periodic domain are revisited. The selection of the correct microcanonical ensemble for the system is first investigated. The numerical results of Weiss and McWilliams [Phys. Fluids A 3, 835 (1991)], who argued that the point vortex system with N=6 is nonergodic because of an apparent discrepancy between ensemble averages and dynamical time averages, are shown to be due to an incorrect ensemble definition. When the correct microcanonical ensemble is sampled, accounting for the vortex momentum constraint, time averages obtained from direct numerical simulation agree with ensemble averages within the sampling error of each calculation, i.e., there is no numerical evidence for nonergodicity. Further, in the N→∞ limit it is shown that the vortex momentum no longer constrains the long-time dynamics and therefore that the correct microcanonical ensemble for statistical mechanics is that associated with the entire constant energy hypersurface in phase space. Next, a recently developed technique is used to generate an explicit formula for the density of states function for the system, including for arbitrary distributions of vortex circulations. Exact formulas for the equilibrium energy spectrum, and for the probability density function of the energy in each Fourier mode, are then obtained. Results are compared with a series of direct numerical simulations with N=50 and excellent agreement is found, confirming the relevance of the results for interpretation of quantum and classical two-dimensional turbulence.

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Phys. Rev. Fluids 2, 014703– Published 11 January, 2017
  • Received 19 August 2016
Article part of CHORUS

©2017 American Physical Society

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