Gravitational instability of isothermal and polytropic spheres

Chavanis, Pierre‐Henri

doi:10.1051/0004-6361:20021779

Cited by 102 publications

(225 citation statements)

References 57 publications

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“…On the other hand, the nature of the caloric curve changes for D = 4 and D = 10. This extends the study performed by Taruya & Sakagami [27,28] and Chavanis [29,26] for D = 3.…”

Section: Introductionsupporting

confidence: 88%

“…Such solutions can result from a process of (possibly incomplete) violent relaxation [26]. Introducing Lagrange multipliers, the first order variations δS − βδE − αδM = 0 lead to…”

Section: Stellar Systemsmentioning

confidence: 99%

“…Since the variational problems (7), (12) and (17) are similar to the usual variational problems that arise in thermodynamics (with the Boltzmann entropy), we can develop a thermodynamical analogy to analyze the dynamical stability of stellar systems, gaseous stars and 2D vortices [26,20,23,17]. In this analogy, the functional S plays the role of a generalized entropy, β is a generalized inverse temperature, β(E) a generalized caloric curve etc...…”

Section: Thermodynamical Analogymentioning

confidence: 99%

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Anomalous diffusion and collapse of self-gravitating Langevin particles inDdimensions

2004

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We address the generalized thermodynamics and the collapse of a system of selfgravitating Langevin particles exhibiting anomalous diffusion in a space of dimension D. This is a basic model of stochastic particles in interaction. The equilibrium states correspond to polytropic configurations similar to stellar polytropes and polytropic stars. 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 SmoluchowskiPoisson 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 of the nonlinear Smoluchowski-Poisson system describing the collapse. Our stability analysis of polytropic spheres can be used to settle the generalized thermodynamical stability of self-gravitating Langevin particles as well as the nonlinear dynamical stability of stellar polytropes, polytropic stars and polytropic vortices.

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“…On the other hand, the nature of the caloric curve changes for D = 4 and D = 10. This extends the study performed by Taruya & Sakagami [27,28] and Chavanis [29,26] for D = 3.…”

Section: Introductionsupporting

confidence: 88%

“…Such solutions can result from a process of (possibly incomplete) violent relaxation [26]. Introducing Lagrange multipliers, the first order variations δS − βδE − αδM = 0 lead to…”

Section: Stellar Systemsmentioning

confidence: 99%

Section: Thermodynamical Analogymentioning

confidence: 99%

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Anomalous diffusion and collapse of self-gravitating Langevin particles inDdimensions

2004

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“…is a particular H-function [18,22,23,28], not an entropy, which occasionally, but not systematically, gives a good fit of the metaequilibrium state. Its maximization at fixed mass and energy leads to a particular class of nonlinearly dynamically stable stationary solutions of the Vlasov equation called stellar polytropes in astrophysics [23].…”

Section: B Vlasov Equation and Violent Relaxationmentioning

confidence: 98%

“…When the two criteria do not coincide, this is similar to a situation of "ensemble inequivalence" in thermodynamics. Such "inequivalence" is observed for the nonlinear dynamical stability of self-gravitating systems such as stellar polytropes and is related to the Antonov first law [22,23]. However, for spatially homogeneous systems, the two criteria are equivalent in general [6] and we shall use here the simpler "canonical" criterion.…”

Section: E Nonlinear Dynamical Stabilitymentioning

confidence: 99%

Hamiltonian and Brownian systems with long-range interactions: II. Kinetic equations and stability analysis

Chavanis¹

2006

Physica A: Statistical Mechanics and its Applications

204

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We discuss the kinetic theory of systems with long-range interactions. We contrast the microcanonical description of an isolated Hamiltonian system described by the Liouville equation from the canonical description of a stochastically forced Brownian system described by the FokkerPlanck equation. We show that the mean-field approximation is exact in a proper thermodynamic limit. For N → +∞, a Hamiltonian system is described by the Vlasov equation. In this collisionless regime, coherent structures can emerge from a process of violent relaxation. These metaequilibrium states, or quasi-stationary states (QSS), are stable stationary solutions of 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. The statistical equilibrium state (Boltzmann) is obtained on a collisional timescale of order N or larger (when the Lenard-Balescu operator cancels out). We also consider the stochastic motion of a test particle in a bath of field particles and derive the general form of the Fokker-Planck equation describing the evolution of the velocity distribution of the test particle. The diffusion coefficient is anisotropic and depends on the velocity of the test particle. For Brownian systems, in the N → +∞ limit, the kinetic equation is a non-local Kramers equation. In the strong friction limit ξ → +∞, or for large times t ≫ ξ −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.

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Statistical Mechanics of Two-Dimensional Vortices and Stellar Systems

Chavanis

2002

Dynamics and Thermodynamics of Systems With Long-Range Interactions

119

181

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The formation of large-scale vortices is an intriguing phenomenon in twodimensional turbulence. Such organization is observed in large-scale oceanic or atmospheric flows, and can be reproduced in laboratory experiments and numerical simulations. A general explanation of this organization was first proposed by Onsager (1949) by considering the statistical mechanics for a set of point vortices in two-dimensional hydrodynamics. Similarly, the structure and the organization of stellar systems (globular clusters, elliptical galaxies,...) in astrophysics can be understood by developing a statistical mechanics for a system of particles in gravitational interaction as initiated by . These statistical mechanics turn out to be relatively similar and present the same difficulties due to the unshielded long-range nature of the interaction. This analogy concerns not only the equilibrium states, i.e. the formation of large-scale structures, but also the relaxation towards equilibrium and the statistics of fluctuations. We will discuss these analogies in detail and also point out the specificities of each system.

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Gravitational instability of isothermal and polytropic spheres

Cited by 102 publications

References 57 publications

Anomalous diffusion and collapse of self-gravitating Langevin particles inDdimensions

Anomalous diffusion and collapse of self-gravitating Langevin particles inDdimensions

Hamiltonian and Brownian systems with long-range interactions: II. Kinetic equations and stability analysis

Statistical Mechanics of Two-Dimensional Vortices and Stellar Systems

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