On the number of unstable modes of an equilibrium

Katz, Joseph

doi:10.1093/mnras/183.4.765

Cited by 189 publications

(190 citation statements)

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“…Our study extends the earlier work of Padmanabhan (1989) in the microcanonical ensemble. By studying the second variations of the free energy, we find that instability sets in precisely at the point of minimum temperature in agreement with the theorem of Katz (1978). The perturbation that induces instability at this point is calculated explicitly; it has not a "core-halo" structure contrary to what happens in the microcanonical ensemble.…”

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confidence: 70%

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Gravitational instability of finite isothermal spheres

Chavanis

2002

A&A

153

341

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Abstract. We investigate the stability of bounded self-gravitating systems in the canonical ensemble by using a thermodynamical approach. Our study extends the earlier work of Padmanabhan (1989) in the microcanonical ensemble. By studying the second variations of the free energy, we find that instability sets in precisely at the point of minimum temperature in agreement with the theorem of Katz (1978). The perturbation that induces instability at this point is calculated explicitly; it has not a "core-halo" structure contrary to what happens in the microcanonical ensemble. We also study Jeans type gravitational instability of isothermal gaseous spheres described by Navier-Stokes equations. The introduction of a container and the consideration of an inhomogeneous distribution of matter avoids the Jeans swindle. We show analytically the equivalence between dynamical stability and thermodynamical stability and the fact that the stability of isothermal gas spheres does not depend on the viscosity. This confirms the findings of Semelin et al. (2001) who used numerical methods or approximations. We also give a simpler derivation of the geometric hierarchy of scales inducing instability discovered by these authors. The density profiles that trigger these instabilities are calculated explicitly; high order modes of instability present numerous oscillations whose nodes also follow a geometric progression. This suggests that the system will fragment in a series of "clumps" and that these "clumps" will themselves fragment in substructures. The fact that both the domain sizes leading to instability and the "clumps" sizes within a domain follow a geometric progression with the same ratio suggests a fractal-like behavior. This gives further support to the interpretation of de Vega et al. (1996) concerning the fractal structure of the interstellar medium.

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confidence: 70%

“…On a theoretical point of view, the stability of isothermal spheres has been first investigated by Katz (1978) with a very powerful method extending Poincaré's theory of linear series of equilibrium. He found that instability sets in precisely at the point of minimum energy.…”

Section: Introductionmentioning

confidence: 99%

Gravitational instability of finite isothermal spheres

Chavanis

2002

A&A

153

341

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“…(36), this minimum value is found to be Ξ = 34.36 (as shown by Antonov 1962). As for the constant {T, V} case, the number of unstable modes for each Ξ coincides with the results of the thermodynamical analysis (see Katz 1978). Once a value of Ξ is obtained, the value of a/b and then an analytical solution G EV is determined.…”

Section: And 4)supporting

confidence: 67%

“…For the present problem this occurs at Ξ ≥ 8.99. When Ξ → ∞, an infinite number of negative eigenvalues appear, precisely at the same points where new unstable modes appear in the thermodynamical approach (see Katz 1978). We now consider the solutions to Eq.…”

Section: Eulerian Representationmentioning

confidence: 99%

Gravothermal catastrophe: The dynamical stability of a fluid model

Sormani

Bertin

2013

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A re-investigation of the gravothermal catastrophe is presented. By means of a linear perturbation analysis, we study the dynamical stability of a spherical self-gravitating isothermal fluid of finite volume and find that the conditions for the onset of the gravothermal catastrophe, under different external conditions, coincide with those obtained from thermodynamical arguments. This suggests that the gravothermal catastrophe may reduce to Jeans instability, rediscovered in an inhomogeneous framework. We find normal modes and frequencies for the fluid system and show that instability develops on the dynamical time scale. We then discuss several related issues. In particular, (1) for perturbations at constant total energy and constant volume, we introduce a simple heuristic term in the energy budget to mimic the role of binaries. (2) We outline the analysis of the two-component case and show how linear perturbation analysis can also be carried out in this more complex context in a relatively straightforward way. (3) We compare the behavior of the fluid model with that of the collisionless sphere. In the collisionless case the instability seems to disappear, which is at variance with the linear Jeans stability analysis in the homogeneous case. We argue that a key ingredient for understanding the difference lies in the role of the detailed angular momentum in a collisionless system. A spherical stellar system is expected to undergo the gravothermal catastrophe only in the presence of some collisionality, which suggests that the instability is dissipative and not dynamical. Finally, we briefly comment on the meaning of the Boltzmann entropy and its applicability to the study of the dynamics of selfgravitating inhomogeneous gaseous systems.

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“…The solutions on the lower branch (points C) have a "core-halo" structure with a degenerate nucleus and a dilute atmosphere; they form the "condensed" phase. According to the theorem of Katz [39], they are both entropy maxima while the intermediate solutions, points B, are unstable saddle points (SP). These points are similar to points A, except that they contain a small embryonic nucleus (with small mass and energy) which plays the role of a "germ" in the langage of phase transitions.…”

Section: Computation Of Fermi-dirac Spheresmentioning

confidence: 99%

Statistical Mechanics of Violent Relaxation in Stellar Systems

Chavanis

2002

Multiscale Problems in Science and Technology

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Abstract. We discuss the statistical mechanics of violent relaxation in stellar systems following the pioneering work of Lynden-Bell (1967). The solutions of the gravitational Vlasov-Poisson system develop finer and finer filaments so that a statistical description is appropriate to smooth out the small-scales and describe the "coarse-grained" dynamics. In a coarse-grained sense, the system is expected to reach an equilibrium state of a Fermi-Dirac type within a few dynamical times. We describe in detail the equilibrium phase diagram and the nature of phase transitions which occur in self-gravitating systems. Then, we introduce a small-scale parametrization of the Vlasov equation and propose a set of relaxation equations for the coarse-grained dynamics. These relaxation equations, of a generalized FokkerPlanck type, are derived from a Maximum Entropy Production Principle (MEPP). We make a link with the quasilinear theory of the Vlasov-Poisson system and derive a truncated model appropriate to collisionless systems subject to tidal forces. With the aid of this kinetic theory, we qualitatively discuss the concept of "incomplete relaxation" and the limitations of Lynden-Bell's theory.

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On the number of unstable modes of an equilibrium

Cited by 189 publications

References 0 publications

Gravitational instability of finite isothermal spheres

Gravitational instability of finite isothermal spheres

Gravothermal catastrophe: The dynamical stability of a fluid model

Statistical Mechanics of Violent Relaxation in Stellar Systems

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