Thermodynamics of a one-dimensional self-gravitating gas with periodic boundary conditions

Abstract

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 cutoff. 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. Numerical simulations of the model, which include the caloric curve, the equation of state, the radial distribution function, and the largest Lyapunov exponent, confirm the existence of the phase transition, and they are in good agreement with the theoretical predictions.

Lyapunov Spectra of Coulombic and Gravitational Periodic Systems

“…For the periodic case studied here with finite N, it turns out that the local peak observed in the largest LCE is the precursor of the phase transition. When plotted versus temperature, this precursor transforms into a cusp characterized by a discontinuity of slope for the gravitational system in the thermodynamics limit [7]. Similar behavior was reported for a self-gravitating ring system [33].…”

“…As experimental techniques progress, it may become possible to control the type of boundary conditions employed in the laboratory. This offers the promise of experimentally observing the phase transition that we predicted in our recent work [7].…”

“…Although of only technical interest for short-range interactions, boundary conditions have a strong influence on the behavior of matter for long-range interactions. In a related work [7], we showed, using both mean field theory as well as the simulation approach presented in this paper, the occurrence of a phase transition in a spatially-periodic, one-dimensional gravitational system. It should be noted that no phase transition occurs in the free-boundary version of the same system.…”

“…In studying possible indicators of phase transitions in small systems, it is useful to know the exact behavior of the system in the thermodynamic limit. One-dimensional systems have proven especially useful in this regard (see [7] for a review of relevant work).…”

Thermodynamics of interacting hard rods on a lattice

“…Hard rods in continuum with gravitational attraction has been investigated in different ensemble in the context of studying gravitational collapse 40,44,45 and it also important for studying effects of gravity on phase transtions in colloids 37 where the spherical symmetry of the gravitational interactions reduce any higher dimensional system to an effective one dimensional system but with structures different from the true one dimensional system. Analytical expression of the partition function and some other thermodynamic quantities have been worked out in 39,40 .…”