A solvable model of hard rods with gravitational interactions

Abstract

We consider a simple 1D model of hard rods with gravitational interactions. First, we consider the situation where the sytem is enclosed in a box with finite size and we exactly compute the equilibrium thermodynamical quantities. Thanks to the confining nature of gravity in 1D which prevents evaporation, the box can be released and we can study an open system with its center of mass fixed. Then, we exactly compute the corresponding equilibrium density profile within the microcanonical ensemble. All those analytical results are discussed in connection with the general issue of ensemble inequivalences for systems with long-range interactions. They also provide specific tests for the reliability of the hydrostatic approach combined with a mean-field treatment of gravitational interactions. In particular, the hydrostatic approach is shown to fail for energies close to the collapse energy where a core-halo structure emerges.