Thermodynamics of the self-gravitating ring model

Abstract

We present the phase diagram, in both the microcanonical and the canonical ensemble, of the self-gravitating-ring (SGR) model, which describes the motion of equal point masses constrained on a ring and subject to 3D gravitational attraction. If the interaction is regularized at short distances by the introduction of a softening parameter, a global entropy maximum always exists, and thermodynamics is well defined in the mean-field limit. However, ensembles are not equivalent and a phase of negative specific heat in the microcanonical ensemble appears in a wide intermediate energy region, if the softening parameter is small enough. The phase transition changes from second to first order at a tricritical point, whose location is not the same in the two ensembles. All these features make of the SGR model the best prototype of a self-gravitating system in one dimension. In order to obtain the stable stationary mass distribution, we apply an iterative method, inspired by a previous one used in 2D turbulence, which ensures entropy increase and, hence, convergence towards an equilibrium state.

Equilibrium and out of equilibrium phase transitions in systems with long range interactions and in 2D flows

“…Such possibilities have first been recognized and studied in the context of self gravitating systems [1, 27,11]. Afterwards, ensemble inequivalence and negative specific heat have been observed or predicted in a number of different physical systems: two dimensional turbulence [12,28,29], plasma physics [28,19], spin systems or toy models [22,25], or self gravitating systems in situations different from the simple initial case [7,5,3,4,6,30,9]. A detailed description of each of these cases is provided in [26].…”

“…We study the phase transitions of this system and how they evolve when ε is varied. Please see [30] for a detailed discussion. Figure 1 shows the caloric curve T (U ) for ε = 10 −2 , where T is the temperature and U is the energy.…”

Phase Transitions in Finite Systems using Information Theory

“…Indeed, equilibrium is a theoretical abstraction which cannot be achieved in the real world 5 . To answer this question we should define a distance.…”

“…Such systems are now studied in many fields of physics, from Bose condensates [38,39] to the quark-gluon plasma [40,41], from cluster melting [6,42] to nuclear fragmentation [7]. Moreover, such inequivalences may survive at the infinite size limit for systems involving long range forces such as self-gravitating objects [3,4,5].…”

“…This may be the case of non saturating forces such as the gravitational [2,3,4,5] or the Coulombic forces. The system may be too small, as in the case of clusters and nuclei [6,7,8,9].…”