Density profiles of a self-gravitating lattice gas in one, two, and three dimensions

We consider a lattice gas in spaces of dimensionality $\mathcal{D}=1,2,3$. The particles are subject to a hardcore exclusion interaction and an attractive pair interaction that satisfies Gauss' law as do Newtonian gravity in $\mathcal{D}=3$, a logarithmic potential in $\mathcal{D}=2$, and a distance-independent force in $\mathcal{D}=1$. Under mild additional assumptions regarding symmetry and fluctuations we investigate equilibrium states of self-gravitating material clusters, in particular radial density profiles for closed and open systems. We present exact analytic results in several instances and high-precision numerical data in others. The density profile of a cluster with finite mass is found to exhibit exponential decay in $\mathcal{D}=1$ and power-law decay in $\mathcal{D}=2$ with temperature-dependent exponents in both cases. In $\mathcal{D}=2$ the gas evaporates in a continuous transition at a nonzero critical temperature. We describe clusters of infinite mass in $\mathcal{D}=3$ with a density profile consisting of three layers (core, shell, halo) and an algebraic large-distance asymptotic decay. In $\mathcal{D}=3$ a cluster of finite mass can be stabilized at $T>0$ via confinement to a sphere of finite radius. In some parameter regime, the gas thus enclosed undergoes a discontinuous transition between distinct density profiles. For the free energy needed to identify the equilibrium state we introduce a construction of gravitational self-energy that works in all $\mathcal{D}$ for the lattice gas. The decay rate of the density profile of an open cluster is shown to transform via a stretched exponential for $1<\mathcal{D}<2$, whereas it crosses over from one power-law at intermediate distances to a different power-law at larger distances for $2<\mathcal{D}<3$.

Figure 1

Main plot: EOS for the ILG (solid line) and the ICG (dashed line) valid in any $\mathcal{D}$. Inset: Isotherms for the ideal Fermi-Dirac gas in $\mathcal{D}=1,2,3$ (solid lines from top down) with reference values $v_T\doteq {\lambda }_T^{\mathcal{D}}$ and $p_T\doteq k_{\mathrm{B}}T/v_T$ (see Appendix pp5) and ideal Maxwell-Boltzmann gas (dashed line). The curves cross over from a linear, ICG-like behavior at low densities to a power-law behavior, $\sim {(v_T/v)}^{1+2/\mathcal{D}}$, at high densities.

Figure 2

Scaled density, pressure, and potential versus scaled radius of a self-gravitating ILG in $\mathcal{D}=1,2,3$ at $\widehat{T}=0$.

Figure 3

Profiles in $\mathcal{D}=1$ of (a) pressure and (b) density for the self-gravitating ILG cluster at $\widehat{T}=0$ (dashed curve) and $\widehat{T}>0$ (solid curves). (c) ILG density profile at higher $\widehat{T}$ (solid curves) in comparison with the asymptotic ICG profiles Eq. (39) (dashed curves) in a log plot. (d) ICG density profiles Eq. (39) at low $\widehat{T}$.

Figure 4

Profiles in $\mathcal{D}=2$ of (a) pressure and (b) density for the self-gravitating ILG cluster at $\widehat{T}=0$ (dashed curve) and $0<\widehat{T}<{\widehat{T}}_{\mathrm{c}}$ (solid curves). (c) ILG density profiles at $0<\widehat{T}<{\widehat{T}}_{\mathrm{c}}$ (solid curves) in comparison with the power-law asymptotics $\sim {\widehat{r}}^{-2/\widehat{T}}$ (dotted lines) in a log-log plot. (d) ICG density profile Eq. (43) at $\widehat{T}>{\widehat{T}}_{\mathrm{c}}$.

Figure 5

Density profiles in $\mathcal{D}=2$ for the self-gravitating ILG confined to a disk-shaped space of radii $\widehat{R}=20,30,50,100$ at temperatures (a) $\widehat{T}=0.45$ (dashed line), and $\widehat{T}={\widehat{T}}_{\mathrm{c}}=0.5$ (solid lines), and (b) $\widehat{T}=0.55$.

Figure 6

Density profiles within the stated temperature range of the ILG confined to a spherical region of radius (a) $\widehat{R}=3$ and (b) $\widehat{R}=4$. Case (a) has a unique solution for all five values of $\widehat{T}$ whereas case (b) has three solutions for the three intermediate values of $\widehat{T}=0.195,0.20,0.205$. As $\widehat{T}$ is being raised, $\rho \left(0\right)$ decreases (increases) for the solid (dashed) curves.

Figure 7

(a) Guggenheim plot of densities ${\rho }_{\mathrm{s}}\left(0\right)$, ${\rho }_{\mathrm{g}}\left(0\right)$ for coexisting profiles at ${\widehat{T}}_{\mathrm{t}}\left(\widehat{R}\right)$. The insets highlight the approaches to the expected cusp singularities as $\widehat{R}\to \infty$. (b) Line of transition temperatures ${\widehat{T}}_{\mathrm{t}}\left(\widehat{R}\right)$ versus inverse radius ${\widehat{R}}^{-1}$ of confinement ending in a critical point $({\widehat{R}}_{\mathrm{c}}^{-1},{\widehat{T}}_{\mathrm{c}})$.

Figure 8

Scaled density profile versus scaled radius of the confined ICG at ${\overline{T}}_{\mathrm{C}}$ (thick line), and four higher temperatures (thin lines).

Figure 9

One-parameter family of density profiles for an open ILG gas in (a) $\mathcal{D}=1$, (b) $\mathcal{D}=2$, and (c) $\mathcal{D}=3$. The parameter values are ${\rho }_0=0,0.9,0.99,0.999$ in each case. The limiting ICG profile, ${\rho }_0\to 0$, is the one decaying most slowly in $\mathcal{D}=1,2$ and the one reaching asymptotic behavior first in $\mathcal{D}=3$. The dashed line in (c) represents Eq. (57).

Figure 10

Data for the exponent ${\nu }_2\left({\rho }_0\right)$ as inferred from Eq. (C6) for open clusters in $\mathcal{D}=2$.

Figure 11

Density profiles of an open ICG cluster (a) in $\mathcal{D}=3$ over a long range of radius and (b) in $\mathcal{D}=3,4,5$ over a shorter range. The dashed lines represent the asymptote Eq. (59).

Figure 12

Crossover between incipient asymptotics, $\sim {\tilde{r}}^{-4}$, at small to intermediate $\tilde{r}$ and true asymptotics, $\sim {\tilde{r}}^{-2}$, at large $\tilde{r}$ of an open ICG cluster in $2<\mathcal{D}<3$. The dotted line represents the exact result Eq. (54) in $\mathcal{D}=2$ and the dashed line the exact asymptote Eq. (57) in $\mathcal{D}=3$.

Figure 13

Stretched exponential asymptotics Eq. (61) of an open ICG cluster in $1<\mathcal{D}<2$. The dotted line represents the exact result Eq. (50) in $\mathcal{D}=1$. The data in panel (b) are extracted from the asymptotes of curves such as shown in panel (a).

Figure 14

Change in gravitational self-energy $d{U}_{\mathrm{S}}$ calculated as work performed against gravity when a thin layer of mass $dm$ is being translocated from radius $r_1$ to radius $r_2$.