Thermodynamics of a Gravitational Gas with Periodic Boundary Conditions

We study the thermodynamic properties of a one-dimensional gas with one-dimensional gravitational interactions. Periodic boundary conditions are implemented as a modification of the potential consisting of a sum over mirror images (Ewald sum), regularized with an exponential cut-off. As a consequence, each particle carries with it its own background density. Using mean field theory, we show that the system has a phase transition at a critical temperature. Above the critical temperature the gas density is uniform, while below the critical point the system becomes inhomogeneous. Precise, event driven, numerical simulations of the model, which include the caloric curve, equation of state, radial distribution function and largest Lyapunov exponent, confirm the existence of the phase transition, and are in good agreement with the theoretical predictions.