https://doi.org/10.1140/epjb/e2014-40869-x

Regular Article

## Statistical mechanics of two-dimensional point vortices: relaxation equations and strong mixing limit

Laboratoire de Physique Théorique (IRSAMC), CNRS and UPS,
Université de Toulouse, 31062
Toulouse,
France

^{a} e-mail: chavanis@irsamc.ups-tlse.fr

Received:
24
September
2013

Received in final form:
25
February
2014

Published online:
7
April
2014

We complement the literature on the statistical mechanics of point vortices in
two-dimensional hydrodynamics. Using a maximum entropy principle, we determine the
multi-species Boltzmann-Poisson equation and establish a form of Virial theorem. Using a
maximum entropy production principle (MEPP), we derive a set of relaxation equations
towards statistical equilibrium. These relaxation equations can be used as a numerical
algorithm to compute the maximum entropy state. We mention the analogies with the
Fokker-Planck equations derived by Debye and Hückel for electrolytes. We then consider the
limit of strong mixing (or low energy). To leading order, the relationship between the
vorticity and the stream function at equilibrium is linear and the maximization of the
entropy becomes equivalent to the minimization of the enstrophy. This expansion is similar
to the Debye-Hückel approximation for electrolytes, except that the temperature is
negative instead of positive so that the effective interaction between like-sign vortices
is attractive instead of repulsive. This leads to an organization at large scales
presenting geometry-induced phase transitions, instead of Debye shielding. We compare the
results obtained with point vortices to those obtained in the context of the statistical
mechanics of continuous vorticity fields described by the Miller-Robert-Sommeria (MRS)
theory. At linear order, we get the same results but differences appear at the next order.
In particular, the MRS theory predicts a transition between sinh and tanh-like
*ω* − *ψ* relationships depending on the
sign of Ku − 3 (where
Ku is the Kurtosis) while
there is no such transition for point vortices which always show a sinh-like
*ω* − *ψ* relationship. We derive the
form of the relaxation equations in the strong mixing limit and show that the enstrophy
plays the role of a Lyapunov functional.

Key words: Statistical and Nonlinear Physics

*© EDP Sciences, Società Italiana di Fisica, Springer-Verlag,
2014*