Gravothermal Catastrophe and Generalized Entropy of Self-Gravitating Systems

We complete the existing literature on the structure and stability of polytropic gas spheres reported in the classical monograph of Chandrasekhar (1942). For isolated polytropes with index $1<n<5$, we provide a new, alternative, proof that the onset of instability occurs for $n=3$ and we express the perturbation profiles of density and velocity at the point of marginal stability in terms of the Milne variables. Then, we consider the case of polytropes confined within a box of...

In this paper, a quantitative characterization for the evolutionary sequence of stellar self-gravitating system is investigated, focusing on the pre-collapse stage of the long-term dynamical evolution. In particular, we consider the quasi-equilibrium behaviors of the N-body systems in the setup of the so-called Antonov problem, i.e., self-gravitating N-body system confined in an adiabatic wall and try to seek a possible connection with thermostatistics of self-gravitating sys...

We present a stability analysis of the classical ideal gas in a new theory of nonextensive statistics and use the theory to understand the phenomena of negative specific heat in some self-gravitating systems. The stability analysis is made on the basis of the second variation of Tsallis entropy. It is shown that the system is thermodynamically unstable if the nonextensive parameter is q>5/3, which is exactly equivalent to the condition of appearance of the negative specific h...

We examine the thermodynamical properties of a family of partially relaxed, anisotropic stellar systems, derived earlier from the Boltzmann entropy under the assumption that a third quantity Q is conserved in addition to the total energy and the total number of stars. We now show that the family of models conforms to the paradigm of the gravothermal catastrophe, which is expected to occur (in the presence of adequate energy transport mechanisms) when the one-parameter equilib...

We discuss the statistical mechanics of rotating self-gravitating systems by allowing properly for the conservation of angular momentum. We study analytically the case of slowly rotating isothermal spheres by expanding the solutions of the Boltzmann-Poisson equation in a series of Legendre polynomials, adapting the procedure introduced by Chandrasekhar (1933) for distorted polytropes. We show how the classical spiral of Lynden-Bell & Wood (1967) in the temperature-energy plan...

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 th...

This work assembles some basic theoretical elements on thermal equilibrium, stability conditions, and fluctuation theory in self-gravitating systems illustrated with a few examples. Thermodynamics deals with states that have settled down after sufficient time has gone by. Time dependent phenomena are beyond the scope of this paper. While thermodynamics is firmly rooted in statistical physics, equilibrium configurations, stability criteria and the destabilizing effect of fluct...

We do not have a final answer to the question of why galaxies choose a particular internal mass distribution. Here we examine whether the distribution is set by thermodynamic equilibrium (TE). Traditionally, TE is discarded for a number of reasons including the inefficiency of two-body collisions to thermalize the mass distribution in a Hubble time, and the fact that the mass distribution maximizing the classical Boltzmann-Gibbs entropy is unphysical. These arguments are ques...

Boltzmann's principle S(E,N,V...)=ln W(E,N,V...) allows the interpretation of Statistical Mechanics of a closed system as Pseudo-Riemannian geometry in the space of the conserved parameters E,N,V... (the conserved mechanical parameters in the language of Ruppeiner) without invoking the thermodynamic limit. The topology is controlled by the curvature of S(E,N,V...). The most interesting region is the region of (wrong) positive maximum curvature, the region of phase-separation....

In this Letter, we investigate the stability of the statistical equilibrium of spherically symmetric collisionless self-gravitating systems. By calculating the second variation of the entropy, we find that perturbations of the relevant physical quantities should be classified as long- and short-range perturbations, which correspond to the long- and short-range relaxation mechanisms, respectively. We show that the statistical equilibrium states of self-gravitating systems are ...