Mean Field Theory of Spherical Gravitating Systems

Important gaps remain in our understanding of the thermodynamics and statistical physics of self-gravitating systems. Using mean field theory, here we investigate the equilibrium properties of several spherically symmetric model systems confined in a finite domain consisting of either point masses, or rotating mass shells of different dimension. We establish a direct connection between the spherically symmetric equilibrium states of a self-gravitating point mass system and a shell model of dimension 3. We construct the equilibrium density functions by maximizing the entropy subject to the usual constraints of normalization and energy, but we also take into account the constraint on the sum of the squares of the individual angular momenta, which is also an integral of motion for these symmetric systems. Two new statistical ensembles are introduced which incorporate the additional constraint. They are used to investigate the possible occurrence of a phase transition as the defining parameters for each ensemble are altered

Effect of angular momentum on equilibrium properties of a self-gravitating system

The microcanonical properties of a two dimensional system of N classical particles interacting via a smoothed Newtonian potential as a function of the total energy E and the total angular momentum L are discussed. In order to estimate suitable observables a numerical method based on an importance sampling algorithm is presented. The entropy surface shows a negative specific heat region at fixed L for all L. Observables probing the average mass distribution are used to understand the link between thermostatistical properties and the spatial distribution of particles. In order to define a phase in non-extensive system we introduce a more general observable than the one proposed by Gross and Votyakov [Eur. Phys. J. B:15, 115 (2000)]: the sign of the largest eigenvalue of the entropy surface curvature. At large E the gravitational system is in a homogeneous gas phase. At low E there are several collapse phases; at L=0 there is a single cluster phase and for L>0 there are several phases with 2 clusters. All these pure phases are separated by first order phase transition regions. The signal of critical behaviour emerges at different points of the parameter space (E,L). We also discuss the ensemble introduced in a recent pre-print by Klinko & Miller; this ensemble is the canonical analogue of the one at constant energy and constant angular momentum. We show that a huge loss of informations appears if we treat the system as a function of intensive parameters: besides the known non-equivalence at first order phase transitions, there exit in the microcanonical ensemble some values of the temperature and the angular velocity for which the corresponding canonical ensemble does not exist, i.e. the partition sum diverges.Comment: 17 pages, 11 figures, submitted to Phys. Rev.

Metastability, negative specific heat and weak mixing in classical long-range many-rotator system

We perform a molecular dynamical study of the isolated $d=1$ classical Hamiltonian ${\cal H} = {1/2} \sum_{i=1}^N L_i^2 + \sum_{i \ne j} \frac{1-cos(\theta_i-\theta_j)}{r_{ij}^\alpha} ;(\alpha \ge 0)$, known to exhibit a second order phase transition, being disordered for $u \equiv U/N{\tilde N} \ge u_c(\alpha,d)$ and ordered otherwise ($U\equiv$ total energy and ${\tilde N} \equiv \frac{N^{1-\alpha/d}-\alpha/d}{1-\alpha/d}$). We focus on the nonextensive case $\alpha/d \le 1$ and observe that, for $u<u_c$, a basin of attraction exists for the initial conditions for which the system quickly relaxes onto a longstanding metastable state (whose duration presumably diverges with $N$ like ${\tilde N}$) which eventually crosses over to the microcanonical Boltzmann-Gibbs stable state. The temperature associated with the (scaled) average kinetic energy per particle is lower in the metastable state than in the stable one. It is exhibited for the first time that the appropriately scaled maximal Lyapunov exponent $\lambda_{u<u_c}^{max}(metastable) \propto N^{-\kappa_{metastable}} ;(N \to \infty)$, where, for all values of $\alpha/d$, $\kappa_{metastable}$ numerically coincides with {\it one third} of its value for $u>u_c$, hence decreases from 1/9 to zero when $\alpha/d$ increases from zero to unity, remaining zero thereafter. This new and simple {\it connection between anomalies above and below the critical point} reinforces the nonextensive universality scenario.Comment: 9 pages and 4 PS figure