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

We complete 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 H\"uckel 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-H\"uckel 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 \omega-\psi 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 \omega-\psi 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

A parametrization of two-dimensional turbulence based on a maximum entropy production principle with a local conservation of energy

In the context of two-dimensional (2D) turbulence, we apply the maximum entropy production principle (MEPP) by enforcing a local conservation of energy. This leads to an equation for the vorticity distribution that conserves all the Casimirs, the energy, and that increases monotonically the mixing entropy ($H$-theorem). Furthermore, the equation for the coarse-grained vorticity dissipates monotonically all the generalized enstrophies. These equations may provide a parametrization of 2D turbulence. They do not generally relax towards the maximum entropy state. The vorticity current vanishes for any steady state of the 2D Euler equation. Interestingly, the equation for the coarse-grained vorticity obtained from the MEPP turns out to coincide, after some algebraic manipulations, with the one obtained with the anticipated vorticity method. This shows a connection between these two approaches when the conservation of energy is treated locally. Furthermore, the newly derived equation, which incorporates a diffusion term and a drift term, has a nice physical interpretation in terms of a selective decay principle. This gives a new light to both the MEPP and the anticipated vorticity method.Comment: To appear in the special IUTAM issue of Fluid Dynamics Research on Vortex Dynamic

Gravitational phase transitions with an exclusion constraint in position space

We discuss the statistical mechanics of a system of self-gravitating particles with an exclusion constraint in position space in a space of dimension $d$. The exclusion constraint puts an upper bound on the density of the system and can stabilize it against gravitational collapse. We plot the caloric curves giving the temperature as a function of the energy and investigate the nature of phase transitions as a function of the size of the system and of the dimension of space in both microcanonical and canonical ensembles. We consider stable and metastable states and emphasize the importance of the latter for systems with long-range interactions. For $d\le 2$, there is no phase transition. For $d>2$, phase transitions can take place between a "gaseous" phase unaffected by the exclusion constraint and a "condensed" phase dominated by this constraint. The condensed configurations have a core-halo structure made of a "rocky core" surrounded by an "atmosphere", similar to a giant gaseous planet. For large systems there exist microcanonical and canonical first order phase transitions. For intermediate systems, only canonical first order phase transitions are present. For small systems there is no phase transition at all. As a result, the phase diagram exhibits two critical points, one in each ensemble. There also exist a region of negative specific heats and a situation of ensemble inequivalence for sufficiently large systems. By a proper interpretation of the parameters, our results have application for the chemotaxis of bacterial populations in biology described by a generalized Keller-Segel model including an exclusion constraint in position space. They also describe colloids at a fluid interface driven by attractive capillary interactions when there is an excluded volume around the particles. Connexions with two-dimensional turbulence are also mentioned

Derivation of the core mass -- halo mass relation of fermionic and bosonic dark matter halos from an effective thermodynamical model

We consider the possibility that dark matter halos are made of quantum particles such as fermions or bosons in the form of Bose-Einstein condensates. In that case, they generically have a "core-halo" structure with a quantum core that depends on the type of particle considered and a halo that is relatively independent of the dark matter particle and that is similar to the NFW profile of cold dark matter. We model the halo by an isothermal gas with an effective temperature $T$. We then derive the core mass -- halo mass relation $M_c(M_v)$ of dark matter halos from an effective thermodynamical model by extremizing the free energy $F(M_c)$ with respect to the core mass $M_c$. We obtain a general relation that is equivalent to the "velocity dispersion tracing" relation according to which the velocity dispersion in the core $v_c^2\sim GM_c/R_c$ is of the same order as the velocity dispersion in the halo $v_v^2\sim GM_v/r_v$. We provide therefore a justification of this relation from thermodynamical arguments. In the case of fermions, we obtain a relation $M_c\propto M_v^{1/2}$ that agrees with the relation found numerically by Ruffini {\it et al.}. In the case of noninteracting bosons, we obtain a relation $M_c\propto M_v^{1/3}$ that agrees with the relation found numerically by Schive {\it et al.}. In the case of bosons with a repulsive self-interaction in the Thomas-Fermi limit, we predict a relation $M_c\propto M_v^{2/3}$ that still has to be confirmed numerically. We also obtain a general approximate core mass -- halo mass relation that is valid for bosons with arbitrary repulsive or attractive self-interaction. For an attractive self-interaction, we determine the maximum halo mass that can harbor a stable quantum core (dilute axion "star")

Phase transitions between dilute and dense axion stars

We study the nature of phase transitions between dilute and dense axion stars interpreted as self-gravitating Bose-Einstein condensates. We develop a Newtonian model based on the Gross-Pitaevskii-Poisson equations for a complex scalar field with a self-interaction potential $V(|\psi|^2)$ involving an attractive $|\psi|^4$ term and a repulsive $|\psi|^6$ term. Using a Gaussian ansatz for the wave function, we analytically obtain the mass-radius relation of dilute and dense axion stars for arbitrary values of the self-interaction parameter $\lambda\le 0$. We show the existence of a critical point $|\lambda|_c\sim (m/M_P)^2$ above which a first order phase transition takes place. We qualitatively estimate general relativistic corrections on the mass-radius relation of axion stars. For weak self-interactions $|\lambda|<|\lambda|_c$, a system of self-gravitating axions forms a stable dilute axion star below a general relativistic maximum mass $M_{\rm max,GR}^{\rm dilute}\sim M_P^2/m$ and collapses into a black hole above that mass. For strong self-interactions $|\lambda|>|\lambda|_c$, a system of self-gravitating axions forms a stable dilute axion star below a Newtonian maximum mass $M_{\rm max,N}^{\rm dilute}=5.073 M_P/\sqrt{|\lambda|}$, collapses into a dense axion star above that mass, and collapses into a black hole above a general relativistic maximum mass $M_{\rm max,GR}^{\rm dense}\sim \sqrt{|\lambda|}M_P^3/m^2$. Dense axion stars explode below a Newtonian minimum mass $M_{\rm min,N}^{\rm dense}\sim m/\sqrt{|\lambda|}$ and form dilute axion stars of large size or disperse away. We determine the phase diagram of self-gravitating axions and show the existence of a triple point $(|\lambda|_*,M_*/(M_P^2/m))$ separating dilute axion stars, dense axion stars, and black holes. We make numerical applications for QCD axions and ultralight axions