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

Abstract

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 $2<\mathcal{D}<3$.