Kinetic theory of point vortices in two dimensions: analytical results and numerical simulations

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 sort of metastable 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

Linear waves in sheared flows. Lower bound of the vorticity growth and propagation discontinuities in the parameters space

This study provides sufficient conditions for the temporal monotonic decay of enstrophy for two-dimensional perturbations traveling in the incompressible, viscous, plane Poiseuille and Couette flows. Extension of J. L. Synge's procedure (1938) to the initial-value problem allowed us to find the region of the wavenumber-Reynolds number map where the enstrophy of any initial disturbance cannot grow. This region is wider than the kinetic energy's one. We also show that the parameters space is split in two regions with clearly distinct propagation and dispersion properties

(In)finiteness of Spherically Symmetric Static Perfect Fluids

This work is concerned with the finiteness problem for static, spherically symmetric perfect fluids in both Newtonian Gravity and General Relativity. We derive criteria on the barotropic equation of state guaranteeing that the corresponding perfect fluid solutions possess finite/infinite extent. In the Newtonian case, for the large class of monotonic equations of state, and in General Relativity we improve earlier results

Anomalous diffusion and collapse of self-gravitating Langevin particles in D dimensions

We address the generalized thermodynamics and the collapse of a system of self-gravitating Langevin particles exhibiting anomalous diffusion in a space of dimension D. The equilibrium states correspond to polytropic distributions. The index n of the polytrope is related to the exponent of anomalous diffusion. We consider a high-friction limit and reduce the problem to the study of the nonlinear Smoluchowski-Poisson system. We show that the associated Lyapunov functional is the Tsallis free energy. We discuss in detail the equilibrium phase diagram of self-gravitating polytropes as a function of D and n and determine their stability by using turning points arguments and analytical methods. When no equilibrium state exists, we investigate self-similar solutions describing the collapse. These results can be relevant for astrophysical systems, two-dimensional vortices and for the chemotaxis of bacterial populations. Above all, this model constitutes a prototypical dynamical model of systems with long-range interactions which possesses a rich structure and which can be studied in great detail.Comment: Submitted to Phys. Rev.

Relativistic stars with a linear equation of state: analogy with classical isothermal spheres and black holes

We complete our previous investigation concerning the structure and the stability of "isothermal" spheres in general relativity. This concerns objects that are described by a linear equation of state $P=q\epsilon$ so that the pressure is proportional to the energy density. In the Newtonian limit $q\to 0$, this returns the classical isothermal equation of state. We consider specifically a self-gravitating radiation (q=1/3), the core of neutron stars (q=1/3) and a gas of baryons interacting through a vector meson field (q=1). We study how the thermodynamical parameters scale with the size of the object and find unusual behaviours due to the non-extensivity of the system. We compare these scaling laws with the area scaling of the black hole entropy. We also determine the domain of validity of these scaling laws by calculating the critical radius above which relativistic stars described by a linear equation of state become dynamically unstable. For photon stars, we show that the criteria of dynamical and thermodynamical stability coincide. Considering finite spheres, we find that the mass and entropy as a function of the central density present damped oscillations. We give the critical value of the central density, corresponding to the first mass peak, above which the series of equilibria becomes unstable. Finally, we extend our results to d-dimensional spheres. We show that the oscillations of mass versus central density disappear above a critical dimension d_{crit}(q). For Newtonian isothermal stars (q=0) we recover the critical dimension d_{crit}=10. For the stiffest stars (q=1) we find d_{crit}=9 and for a self-gravitating radiation (q=1/d) we find d_{crit}=9.96404372... very close to 10. Finally, we give analytical solutions of relativistic isothermal spheres in 2D gravity.Comment: A minor mistake in calculation has been corrected in the second version (v2

Long-range thermoelectric effects in mesoscopic superconductor-normal metal structures

We consider a mesoscopic four-terminal superconductor/normal metal (S/N) structure in the presence of a temperature gradient along the N wire. A thermoemf arises in this system even in the absence of the thermoelectric quasiparticle current if the phase difference between the superconductors is not zero. We show that the thermoemf is not small in the case of a negligible Josephson coupling between two superconductors. It is also shown that the thermoelectric voltage has two maxima: one at a low temperature and another at a temperature close to the critical temperature. The obtained temperature dependence of the thermoemf describes qualitatively experimental data.Comment: 9 pages, 6 figure

Brownian theory of 2D turbulence and generalized thermodynamics

We propose a new parametrization of 2D turbulence based on generalized thermodynamics and Brownian theory. Explicit relaxation equations are obtained that should be easily implementable in numerical simulations for three typical types of turbulent flows. Our parametrization is related to previous ones but it removes their defects and offers attractive new perspectives.Comment: Submitted to Phys. Rev. Let

Hamiltonian and Brownian systems with long-range interactions

We discuss the dynamics and thermodynamics of systems with long-range interactions. We contrast the microcanonical description of an isolated Hamiltonian system to the canonical description of a stochastically forced Brownian system. We show that the mean-field approximation is exact in a proper thermodynamic limit. The equilibrium distribution function is solution of an integrodifferential equation obtained from a static BBGKY-like hierarchy. It also optimizes a thermodynamical potential (entropy or free energy) under appropriate constraints. We discuss the kinetic theory of these systems. In the $N\to +\infty$ limit, a Hamiltonian system is described by the Vlasov equation. To order 1/N, the collision term of a homogeneous system has the form of the Lenard-Balescu operator. It reduces to the Landau operator when collective effects are neglected. We also consider the motion of a test particle in a bath of field particles and derive the general form of the Fokker-Planck equation. The diffusion coefficient is anisotropic and depends on the velocity of the test particle. This can lead to anomalous diffusion. For Brownian systems, in the $N\to +\infty$ limit, the kinetic equation is a non-local Kramers equation. In the strong friction limit $\xi\to +\infty$, or for large times $t\gg \xi^{-1}$, it reduces to a non-local Smoluchowski equation. We give explicit results for self-gravitating systems, two-dimensional vortices and for the HMF model. We also introduce a generalized class of stochastic processes and derive the corresponding generalized Fokker-Planck equations. We discuss how a notion of generalized thermodynamics can emerge in complex systems displaying anomalous diffusion.Comment: The original paper has been split in two parts with some new material and correction

On quadratic integral equations in Orlicz spaces

AbstractIn this paper we study the quadratic integral equation of the formx(t)=g(t)+λx(t)∫abK(t,s)f(s,x(s))ds. Several existence theorems for a.e. monotonic solutions in Orlicz spaces are proved for strongly nonlinear functions f. The presented method of the proof can be easily extended to different classes of solutions