Phase transitions in self-gravitating systems: Self-gravitating fermions and hard-sphere models

Abstract

We discuss the nature of phase transitions in self-gravitating systems both in the microcanonical and in the canonical ensemble. We avoid the divergence of the gravitational potential at short distances by considering the case of self-gravitating fermions and hard spheres models. Depending on the values of the parameters, three kinds of phase transitions (of zeroth, first and second order) are evidenced. They separate a "gaseous" phase with a smoothly varying distribution of matter from a "condensed" phase with a core-halo structure. We propose a simple analytical model to describe these phase transitions. We determine the value of energy (in the microcanonical ensemble) and temperature (in the canonical ensemble) at the transition point and we study their dependance with the degeneracy parameter (for fermions) or with the size of the particles (for a hard spheres gas). Scaling laws are obtained analytically in the asymptotic limit of a small short distance cut-off. Our analytical model captures the essential physics of the problem and compares remarkably well with the full numerical solutions. We also stress some analogies with the liquid/gas transition and with the Blume-Emery-Griffiths (BEG) model with infinite range interactions. In particular, our system presents two tricritical points at which the transition passes from first to second order.

Statistical Mechanics of Violent Relaxation in Stellar Systems

“…A simple analytical model of these phase transitions has been proposed in Ref. [20] and provides a fairly good agreement with the full numerical solution. A particularity of self-gravitating systems, which are in essence non-extensive, is that the statistical ensembles (microcanonical and canonical) are not interchangeable.…”

“…Therefore, the description of the equilibrium diagram is different whether the system evolves at fixed energy of fixed temperature. A discussion of this interesting phenomenon can be found in the review of Padmanabhan [48] and in Chavanis [20]. For astrophysical purposes, it is still a matter of debate to decide whether collisionless stellar systems like elliptical galaxies are degenerate (in the sense of Lynden-Bell) or not.…”

Historical and Physical Account on Entropy and Perspectives on the Second Law of Thermodynamics for Astrophysical and Cosmological Systems

“…, ln 3 2 ln 2 3 ln (27) Let us simplify the analysis. In the beginning of the collapsing process, our cloud is diffuse, i.e., the particles are far apart from each other.…”

“…It is noteworthy that after Hawking published his manuscript about what is now called "Hawking radiation", the expression for T bh has been considered (even by Bekenstein) the actual temperature of the black hole, despite Bekenstein's disclaimer in his prior to 1975 manuscripts. Another questionable procedure is the usage of canonical ensemble in astrophysical systems [23,25,27]. It is established that the canonical ensemble presupposes thermal equilibrium between the system of interest and a much larger system (a heat reservoir).…”

“…The inclusion an insulating box in the black hole system evidences a paradigmatic construction without an effective tie to an actual physical system. Note that placing a system inside a box of volume V is a standard and well accepted procedure in astrophysics [20,[23][24][25][26][27].…”