Thermodynamics of the self-gravitating ring model

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.

Figure 1

Self-gravitating ring model with a fixed unitary radius. Particles are constrained to move on a ring and therefore their location is specified by the angles measured with respect to a fixed direction. Each pair of particles at ${\theta }_i$ and ${\theta }_j$ interacts through the inverse-square three-dimensional gravitational force. The distance is measured by the chord, as shown in the figure.

Figure 2

Caloric curves of the self-gravitating ring (SGR) model obtained from numerical simulations of Hamiltonian (1). Panel (a) refers to the softening parameter value $\epsilon =10$, for which a second order phase transition appears at $U_c$. No backbending of the caloric curve, indicating a negative specific heat, is present. Simulations were performed for $N=100$. Panel (b) shows the caloric curves for two different values of the softening parameter, ${\epsilon }_1=1.0\times {10}^{-6}$ and ${\epsilon }_2=2.5\times {10}^{-7}$, and $N=100$. The transition is here first order in the microcanonical ensemble (see Sec. 6 for a discussion). The two transition energies $U_c\left({\epsilon }_1\right)$ and $U_c\left({\epsilon }_2\right)$ are pretty close, suggesting a slow variation of the critical energy with the softening parameter $\epsilon$. On the contrary, $U_{\mathit{\text{top}}}\left({\epsilon }_1\right)$ is significantly smaller than $U_{\mathit{\text{top}}}\left({\epsilon }_2\right)$, indicating that this characteristic energy value diminishes with $\epsilon$. A negative specific heat phase appears for $U_{\mathit{\text{top}}}<U<U_c$, and expands as the softening parameter is reduced. All quantities are plotted in arbitrary units.

Figure 3

Convergence of the entropy using the algorithm of Sec. 5 for $\epsilon ={10}^{-5}$ and $U=-1$. All quantities are plotted in arbitrary units.

Figure 4

Temperature [panel (a)] and entropy [panel (b)] versus energy $U$ for the softening parameter value $\epsilon ={10}^{-5}$. Four values of the energy, indicated by the short-dashed vertical lines, can be identified from this picture: $U_{\mathit{\text{low}}}\simeq -93$ and $U_{\mathit{\text{high}}}\simeq 6$ bound from below and above the region of inequivalence of ensembles. $U_c\simeq 0$ is the transition energy in the microcanonical ensemble. $U_{\mathit{\text{top}}}\simeq -66$ limits from below the negative specific heat region, where temperature decreases as energy increases. $T_{\mathit{can}}\simeq 15$, represented with a dashed line in panel (a), is the canonical transition temperature and corresponds to the inverse slope of the entropy, both at $U_{\mathit{\text{low}}}$ and $U_{\mathit{\text{high}}}$, as represented by the straight dashed line in panel (b). The full lines represent the analytical solutions of the temperature and of entropy in the uniform case [see formulas (43, 44)]. They are extended slightly below $U_c$, in the metastable phase, in order to identify them. The insets in panels (a) and (b) show a zoom of the temperature and of the entropy around $U_c$, revealing a temperature jump at $U_c$ and different slopes of the entropy above and below $U_c$, which emphasizes the first order nature of the phase transition. All quantities are plotted in arbitrary units.

Figure 5

“Magnetization” $B$ versus energy $U$ for $\epsilon ={10}^{-5}$, which emphasizes the microcanonical first order phase transition at $U_c\simeq 0$ by showing a jump in the order parameter. All quantities are plotted in arbitrary units.

Figure 6

A typical mass density distribution $\rho \left(\theta \right)$ for $\epsilon ={10}^{-5}$ and $U=-20.0$, in the negative specific heat region. All quantities are plotted in arbitrary units.

Figure 7

Both the high energy branch of the entropy versus energy curve, corresponding to the homogeneous solution (solid line) and the low energy branch of the inhomogeneous solution (dashed line) are represented in this plot for $\epsilon ={10}^{-5}$. The two branches cross at $U_c\simeq 0$. The continuation of the homogeneous branch into the low energy region is bounded from below by $U_{\mathit{hom}}\simeq -1.19$, indicated by a vertical dashed line. The inhomogeneous branch continues into the high energy phase and ends at $U_{\mathit{in}}\simeq 0.16$, again indicated by a vertical dashed line. All quantities are plotted in arbitrary units.

Figure 8

Panel (a): The caloric curve (triangles) for $\epsilon ={10}^{-2}$. The dashed vertical lines indicate, from left to right, $U_{\mathit{\text{low}}}\simeq -1.98$, $U_{\mathit{\text{top}}}\simeq -1.3$, $U_c\simeq -0.32$, and $U_{\mathit{\text{high}}}\simeq -0.225$. The homogeneous phase curve, known analytically, is shown by the continuous line. The main difference with respect to Fig. 4 is that now there is not a temperature jump at $U_c$. The phase transition is second order in the microcanonical ensemble, while it is still first order in the canonical ensemble, at $T_{\mathit{can}}\simeq 0.8$. Panel (b): Entropy versus energy (triangles) for $\epsilon ={10}^{-2}$. The entropy curve corresponding to the inhomogeneous distribution smoothly connects with the one of the homogeneous distribution (solid line). The oblique straight dashed line is tangent to the entropy at $U_{\mathit{\text{low}}}$ and $U_{\mathit{\text{high}}}$, which delimit the energy region of ensemble inequivalence. All quantities are plotted in arbitrary units.

Figure 9

“Magnetization” $B$ versus energy $U$ for $\epsilon ={10}^{-2}$ which emphasizes the microcanonical second order phase transition at $U_c$, because the order parameter vanishes continuously. All quantities are plotted in arbitrary units.

Figure 10

The dash-dotted line represents $U_{\mathit{in}}\left(\epsilon \right)$, the dashed line $U_{\mathit{hom}}={\overline{E}}_p\left(\epsilon \right)$, the filled circles the first order microcanonical phase transition energy, and the open circles the second order one. At the microcanonical tricritical point ${\epsilon }_T^{\mu }\simeq {10}^{-4}$, the phase transition changes from first order to second order and, at the same time, the inhomogeneous metastable solution disappears. All quantities are plotted in arbitrary units.

Figure 11

$\epsilon$ dependence of $U_{\mathit{\text{low}}}$ (dashed line), $U_{\mathit{high}}$ (solid line), and $U_{\mathit{\text{top}}}$ (dash-dotted line). The canonical tricritical point is located at ${\epsilon }_T^c\simeq {10}^{-1}$ where the three curves merge. At this softening parameter value also the negative specific heat disappears in the microcanonical ensemble, while the transition becomes second order in the canonical ensemble. In the figure, we also show, with a long-dashed line, the theoretical estimate of $U_{\mathit{\text{top}}}^{\mathit{th}}\simeq -1∕\left(4\sqrt{2\epsilon }\right)$ obtained in Ref. 14. All quantities are plotted in arbitrary units.

Figure 12

Comparison of the mass density profile obtained by the iterative method (solid line) with the result of numerical simulations (plus signs) with $N=4000$ and $\epsilon ={10}^{-5}$. Parameter values are the same as Fig. 6. The inset is a zoom of the center of the profile. All quantities are plotted in arbitrary units.

Figure 13

Time evolution of the virial ratio $\mid 2K∕V\mid$ of the SGR model for $\epsilon ={10}^{-5}$ and $N=4000$. Initially, the particles are homogeneously distributed in the interval $[0,2\pi ∕75]$ with zero kinetic energy. The virial ratio oscillates asymptotically around the value 0.49, which differs significantly from the equilibrium value 0.55 indicated by the dashed horizontal line. The initial virial ratio is zero, although this time region is not visible in the figure. All quantities are plotted in arbitrary units.

Figure 14

Relaxation to different maximum entropy states in the SGR model for $U\simeq 0$, $\epsilon ={10}^{-5}$, and $N={10}^3$. Panel (a) shows the relaxation of the temperature either to the inhomogeneous state value (horizontal dotted line), or to the homogeneous one (horizontal dash-dotted line), depending on whether the particles are initially distributed on a smaller arch $\theta ∊[0,\pi ∕50]$ (solid line) or a larger arch $\theta ∊[0,\pi ∕5]$ (dashed line). In both cases, the velocity distribution is initially a “water bag.” Panel (b) shows the same for the “magnetization.” All quantities are plotted in arbitrary units.