Kinetic theory of point vortices in two dimensions: analytical results and numerical simulations |
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Authors: | P H Chavanis M Lemou |
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Institution: | (1) Laboratoire de Physique Théorique (CNRS UMR 5152), Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France;(2) Mathématiques pour l'Industrie et la Physique (CNRS UMR 5640), Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France |
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Abstract: | We develop the kinetic theory of point vortices in two-dimensional
hydrodynamics and illustrate the main results of the
theory with numerical simulations. We first consider the evolution of
the system “as a whole” and show that the evolution of the
vorticity profile is due to resonances between different orbits of the
point vortices. The evolution stops when the profile of angular
velocity becomes monotonic even if the system has not reached the
statistical equilibrium state (Boltzmann distribution). In that case,
the system remains blocked in a quasi stationary state with a non
standard distribution. We also study the relaxation of a test vortex
in a steady bath of field vortices. The relaxation of the test vortex
is described by a Fokker-Planck equation involving a diffusion term
and a drift term. The diffusion coefficient, which is proportional to
the density of field vortices and inversely proportional to the shear,
usually decreases rapidly with the distance. The drift is proportional
to the gradient of the density profile of the field vortices and is
connected to the diffusion coefficient by a generalized Einstein
relation. We study the evolution of the tail of the distribution
function of the test vortex and show that it has a front structure. We
also study how the temporal auto-correlation function of the position
of the test vortex decreases with time and find that it usually
exhibits an algebraic behavior with an exponent that we compute
analytically. We mention analogies with other systems with long-range
interactions. |
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Keywords: | 05 20 -y Classical statistical mechanics 05 45 -a Nonlinear dynamics and nonlinear dynamical systems 05 20 Dd Kinetic theory 47 10 -g General theory in fluid dynamics 47 32 C- Vortex dynamics |
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