Out-of-equilibrium behavior of quantum vortices: A comparison of point vortex dynamics and Fokker-Planck evolution

An out-of-equilibrium two-dimensional superfluid relaxes towards equilibrium via a process of coarsening, driven by the annihilation of vortices with antivortices. Here we present a comparison of two different numerical models of this process, a dissipative point vortex model and Fokker-Planck evolution, across a wide range of initial configurations and levels of dissipation. We find that our dissipative point vortex model is well approximated by Fokker-Planck evolution only for very low initial energies per vortex, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:msup><a:mi>E</a:mi><a:mn>0</a:mn></a:msup><a:mo>/</a:mo><a:msub><a:mi>N</a:mi><a:mi>v</a:mi></a:msub><a:mo>≲</a:mo><a:mo>−</a:mo><a:mn>4</a:mn></a:mrow></a:math>, when almost all vortices and antivortices are closely bound into dipoles. We observe that the dynamical critical exponent <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>z</b:mi></b:math> in the dissipative point vortex model undergoes a crossover, from a roughly constant value close to 2 for <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:msup><c:mi>E</c:mi><c:mn>0</c:mn></c:msup><c:mo>/</c:mo><c:msub><c:mi>N</c:mi><c:mi>v</c:mi></c:msub><c:mo>≳</c:mo><c:mo>−</c:mo><c:mn>4</c:mn></c:mrow></c:math>, to a value which depends explicitly on the initial conditions for <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:msup><d:mi>E</d:mi><d:mn>0</d:mn></d:msup><d:mo>/</d:mo><d:msub><d:mi>N</d:mi><d:mi>v</d:mi></d:msub><d:mo>≲</d:mo><d:mo>−</d:mo><d:mn>5</d:mn></d:mrow></d:math>.