arXiv subjects

Pierre-Henri Chavanis

Publications and source records attributed to Pierre-Henri Chavanis.

176 records · Page 10Linked to original sources

arXiv2004

Chapman-Enskog derivation of the generalized Smoluchowski equation

We use the Chapman-Enskog method to derive the Smoluchowski equation from the Kramers equation in a high friction limit. We consider two main extensions of this problem: we take into account a uniform rotation of the background medium and we consider a generalized class of Kramers equations associated with generalized free energy functionals. We mention applications of these results to systems with long-range interactions (self-gravitating systems, 2D vortices, bacterial populations,...). In that case, the Smoluchowski equation is non-local. In the limit of short-range interactions, it reduces to a generalized form of the Cahn-Hilliard equation. These equations are associated with an effective generalized thermodynamical formalism.

cond-mat.stat-mech↗

arXiv2004

Post-collapse dynamics of self-gravitating Brownian particles in D dimensions

We address the post-collapse dynamics of a self-gravitating gas of Brownian particles in D dimensions, in both canonical and microcanonical ensembles. In the canonical ensemble, the post-collapse evolution is marked by the formation of a Dirac peak with increasing mass. The density profile outside the peak evolves self-similarly with decreasing central density and increasing core radius. In the microcanonical ensemble, the post-collapse regime is marked by the formation of a ``binary''-like structure surrounded by an almost uniform halo with high temperature. These results are consistent with thermodynamical predictions.

cond-mat.stat-mech↗

arXiv2004

Estimate of blow-up and relaxation time for self-gravitating Brownian particles and bacterial populations

We determine an asymptotic expression of the blow-up time t_coll for self-gravitating Brownian particles or bacterial populations (chemotaxis) close to the critical point. We show that t_coll=t_{*}(eta-eta_c)^{-1/2} with t_{*}=0.91767702..., where eta represents the inverse temperature (for Brownian particles) or the mass (for bacterial colonies), and eta_c is the critical value of eta above which the system blows up. This result is in perfect agreement with the numerical solution of the Smoluchowski-Poisson system. We also determine the asymptotic expression of the relaxation time close but above the critical temperature and derive a large time asymptotic expansion for the density profile exactly at the critical point.

cond-mat.stat-mech↗

arXiv2003

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.

cond-mat.stat-mech↗

arXiv2003

Generalized thermodynamics and Fokker-Planck equations. Applications to stellar dynamics, two-dimensional turbulence and Jupiter's great red spot

We introduce a new set of generalized Fokker-Planck equations that conserve energy and mass and increase a generalized entropy until a maximum entropy state is reached. The concept of generalized entropies is rigorously justified for continuous Hamiltonian systems undergoing violent relaxation. Tsallis entropies are just a special case of this generalized thermodynamics. Application of these results to stellar dynamics, vortex dynamics and Jupiter's great red spot are proposed. Our prime result is a novel relaxation equation that should offer an easily implementable parametrization of geophysical turbulence. This relaxation equation depends on a single key parameter related to the skewness of the fine-grained vorticity distribution. Usual parametrizations (including a single turbulent viscosity) correspond to the infinite temperature limit of our model. They forget a fundamental systematic drift that acts against diffusion as in Brownian theory. Our generalized Fokker-Planck equations may have applications in other fields of physics such as chemotaxis for bacterial populations. We propose the idea of a classification of generalized entropies in classes of equivalence and provide an aesthetic connexion between topics (vortices, stars, bacteries,...) which were previously disconnected.

cond-mat.stat-mech↗

arXiv2003

Generalized thermodynamics and kinetic equations: Boltzmann, Landau, Kramers and Smoluchowski

We propose a formal extension of thermodynamics and kinetic theories to a larger class of entropy functionals. Kinetic equations associated to Boltzmann, Fermi, Bose and Tsallis entropies are recovered as a special case. This formalism first provides a unifying description of classical and quantum kinetic theories. On the other hand, a generalized thermodynamical framework is justified to describe complex systems exhibiting anomalous diffusion. Finally, a notion of generalized thermodynamics emerges in the context of the the violent relaxation of collisionless stellar systems and two-dimensional vortices due to the existence of Casimir invariants and incomplete relaxation. A thermodynamical analogy can also be developed to analyze the nonlinear dynamical stability of stationary solutions of the Vlasov and 2D Euler-Poisson systems. On general grounds, we suggest that generalized entropies arise due to the existence of ``hidden constraints'' that modify the form of entropy that we would naively expect. Generalized kinetic equations are therefore ``effective'' equations that are introduced heuristically to describe complex systems.

cond-mat.stat-mech↗

arXiv2001

On the effective velocity created by a point vortex in two-dimensional hydrodynamics

We complete previous investigations on the statistics of velocity fluctuations arising from a random distribution of point vortices in two-dimensional hydrodynamics. We show that, on a statistical sense, the velocity created by a point vortex is shielded by cooperative effects on a distance $Λ\sim n^{-1/2}$, the inter-vortex separation. For $R\gg Λ$, the ``effective'' velocity decays as $R^{-2}$ instead of the ordinary law $R^{-1}$ recovered for $R\ll Λ$. These results are similar to those obtained by Agekyan [Sov. Astron. 5 (1962) 809] in his investigations on the fluctuations of the gravitational field. They give further support to our previous observation that the statistics of velocity fluctuations are (marginally) dominated by the contribution of the nearest neighbor.

cond-mat.stat-mech↗

arXiv2001

Gravitational instability of finite isothermal spheres in general relativity. Analogy with neutron stars

We investigate the effects of relativity on the gravitational instability of finite isothermal gaseous spheres. In the first part of the paper, we treat the gravitational field within the framework of Newtonian mechanics but we use a relativistic equation of state in the condition of hydrostatic equilibrium. In the second part of the paper, we study the full general relativistic problem for a gas described by an equation of state $p=qε$ such that the pressure is proportional to the energy density (``isothermal'' distribution). For $q=1/3$, this equation of state describes the core of neutron stars. The mass-density diagram displays some damped oscillations and there exists a critical value of mass-energy above which no equilibrium state is possible. We show analytically that the mass peaks are associated with new modes of instability. These results are strikingly similar to those obtained by Antonov [Vest. Leningr. Gos. Univ. 7, 135 (1962)] and Lynden-Bell & Wood (1968) for a classical isothermal gas. Our study completes the analogy between isothermal spheres and neutron stars investigated by Yabushita [MNRAS 167, 95 (1974)].

astro-ph↗

arXiv2001

The spatial correlations in the velocities arising from a random distribution of point vortices

This paper is devoted to a statistical analysis of the velocity fluctuations arising from a random distribution of point vortices in two-dimensional turbulence. Exact results are derived for the correlations in the velocities occurring at two points separated by an arbitrary distance. We find that the spatial correlation function decays extremely slowly with the distance. We discuss the analogy with the statistics of the gravitational field in stellar systems.

cond-mat.stat-mech↗

arXiv2000

Quasilinear theory of the 2D Euler equation

We develop a quasilinear theory of the 2D Euler equation and derive an integro-differential equation for the evolution of the coarse-grained vorticity. This equation respects all the invariance properties of the Euler equation and conserves angular momentum in a circular domain and linear impulse in a channel. We show under which hypothesis we can derive a H-theorem for the Fermi-Dirac entropy and make the connection with statistical theories of 2D turbulence.

physics.flu-dyn↗

arXiv2000

Scaling laws and vortex profiles in 2D decaying turbulence

We use high resolution numerical simulations over several hundred of turnover times to study the influence of small scale dissipation onto vortex statistics in 2D decaying turbulence. A self-similar scaling regime is detected when the scaling laws are expressed in units of mean vorticity and integral scale, as predicted by Carnevale et al., and it is observed that viscous effects spoil this scaling regime. This scaling regime shows some trends toward that of the Kirchhoff model, for which a recent theory predicts a decay exponent $ξ=1$. In terms of scaled variables, the vortices have a similar profile close to a Fermi-Dirac distribution.

cond-mat.stat-mech↗

arXiv2000

Numerical renormalization group of vortex aggregation in 2D decaying turbulence: the role of three-body interactions

In this paper, we introduce a numerical renormalization group procedure which permits long-time simulations of vortex dynamics and coalescence in a 2D turbulent decaying fluid. The number of vortices decreases as $N\sim t^{-ξ}$, with $ξ\approx 1$ instead of the value $ξ=4/3$ predicted by a na\"ıve kinetic theory. For short time, we find an effective exponent $ξ\approx 0.7$ consistent with previous simulations and experiments. We show that the mean square displacement of surviving vortices grows as $ \sim t^{1+ξ/2}$. Introducing effective dynamics for two-body and three-body collisions, we justify that only the latter become relevant at small vortex area coverage. A kinetic theory consistent with this mechanism leads to $ξ=1$. We find that the theoretical relations between kinetic parameters are all in good agreement with experiments.

cond-mat.stat-mech↗

arXiv2000

The statistics of velocity fluctuations arising from a random distribution of point vortices: the speed of fluctuations and the diffusion coefficient

This paper is devoted to a statistical analysis of the fluctuations of velocity and acceleration produced by a random distribution of point vortices in two-dimensional turbulence. We show that the velocity probability density function (p.d.f.) behaves in a manner which is intermediate between Gaussian and Lévy laws while the distribution of accelerations is governed by a Cauchy law. Our study accounts properly for a spectrum of circulations among the vortices. In the case of real vortices (with a finite core) we show analytically that the distribution of accelerations makes a smooth transition from Cauchy (for small fluctuations) to Gaussian (for large fluctuations) passing probably through an exponential tail. We introduce a function $T(V)$ which gives the typical duration of a velocity fluctuation $V$; we show that $T(V)$ behaves like $V$ and $V^{-1}$ for weak and large velocities respectively. These results have a simple physical interpretation in the nearest neighbor approximation and in Smoluchowski theory concerning the persistence of fluctuations. We discuss the analogies with respect to the fluctuations of the gravitational field in stellar systems. As an application of these results, we determine an approximate expression for the diffusion coefficient of point vortices. When applied to the context of freely decaying two-dimensional turbulence, the diffusion becomes anomalous and we establish a relationship $ν=1+{ξ\over 2}$ between the exponent of anomalous diffusion $ν$ and the exponent $ξ$ which characterizes the decay of the vortex density. This relation is in good agreement with laboratory experiments and numerical simulations.

cond-mat.stat-mech↗

arXiv1999

Trapping of dust by coherent vortices in the solar nebula

We develop the idea proposed by Barge & Sommeria (1995) and Tanga et al. (1996) that large-scale vortices present in the solar nebula can concentrate dust particles and facilitate the formation of planetesimals and planets. We introduce an exact vortex solution of the incompressible 2D Euler equation and study the motion of dust particles in that vortex. In particular, we derive analytical expressions for the capture time and the mass capture rate as a function of the friction parameter. Then, we study how small-scale turbulent fluctuations affect the motion of the particles in the vortex and determine their rate of escape by solving a problem of quantum mechanics. We apply these results to the solar nebula and find that the capture is optimum near Jupiter's orbit (as noticed already by Barge & Sommeria 1995) but also in the Earth region. This second optimum corresponds to the transition between the Epstein and the Stokes regime which takes place, for relevant particles, at the separation between telluric and giant planets (i.e near the asteroid belt). At these locations, the particles are efficiently captured and concentrated by the vortices and can undergo gravitational collapse to form the planetesimals.

astro-ph↗