  We study the thermodynamical properties of a one-dimensional gas with
one-dimensional gravitational interactions, and placed in a uniform mass
background. 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. 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. Numerical
simulations of the model confirms the existence of the phase transition, and
are in good agreement with the theoretical results.
