Recent progress in the study of one-dimensional gravitating systems.

We review recent developments in the study of two different interacting gravitational systems: the system of parallel planar mass sheets and the system of concentric spherical mass shells. The approach to equilibrium of a system of parallel planar mass sheets is investigated. Parallels with three-dimensional systems are described. Mass segregation and kinetic energy equipartition in a two-component system of planar mass sheets is demonstrated via numerical simulation. The existence of two distinct phases is demonstrated in the system of spherical mass shells. The nature of the transition in the microcanonical, canonical, and grand canonical ensembles is studied both theoretically in terms of mean-field theory and via dynamical simulation.

Path-integral study of positronium decay in xenon.

We use path-integral Monte Carlo to study the properties of a quantum particle equilibrated in a classical Lennard-Jones fluid. By choosing 2m(e) for its mass, and potential parameters corresponding to xenon, we are able to model the behavior of thermalized positronium above the xenon critical temperature. We carefully study the local distortion of the fluid in the neighborhood of the quantum particle, and use this information to compute the annihilation rate as a function of density on two isotherms. The results compare favorably with experiment below the critical point density. Contrary to accepted views, we demonstrate that positronium remains in a self-trapped state at over twice the critical point density.

Rotation-induced phase transition in a spherical gravitating system.

Due to the infinite range and singularity of the gravitational force, it is difficult to directly apply the standard methods of statistical physics to self-gravitating systems, e.g., interstellar grains, globular clusters, galaxies, etc. Unusual phenomena can occur, such as a negative heat capacity, unbounded mass, or the gravothermal catastrophe where the equilibrium state is fully collapsed and the entropy is unbounded. Using mean field theory, we investigate the influence of rotation on a purely spherical gravitational system. Although spherical symmetry nullifies the total angular momentum, its square is finite and conserved. Here we study the case where each particle has specific angular momentum of the same magnitude l. We rigorously prove the existence of an upper bound on the entropy and a lower bound for the energy. We demonstrate that, in the microcanonical and canonical ensembles, a phase transition occurs when l falls below a critical value. We characterize the properties of each phase and construct the coexistence curve for each ensemble. Possible applications to astrophysics are considered.

Dynamics in a discontinuous field: The smooth Fermi piston.

In this paper we examine a version of the Fermi piston with a discontinuous, but nonimpulsive, periodic driving force. The dynamics of a particle moving in one spatial dimension are studied using a combination of numerical and analytical techniques. The configuration space of the particle is divided into two regions of constant acceleration that are of equal magnitude and opposite direction. The point of discontinuity F(t) dividing the regions changes periodically in time. The method of surface-of-section is used to study the phase space (phi(n), v(n)), where phi(n) is the phase of the driving function and v(n) is the velocity of the particle at the nth encounter between the particle and boundary. We show that it is not possible to stochastically drive up the energy indefinitely except for the cases where F is discontinuous, or dF/dt is not finite everywhere. In addition, we find a new mechanism, other than KAM tori, for segmenting the phase space. As in the KAM picture, the central cause of the new behavior is resonance between the natural period of the particle and the period of the driving force. The boundaries to diffusion for continuous driving functions result from parabolic fixed points that span the entire phase range.

Dynamics of a gravitational billiard with a hyperbolic lower boundary.

Gravitational billiards provide a simple method for the illustration of the dynamics of Hamiltonian systems. Here we examine a new billiard system with two parameters, which exhibits, in two limiting cases, the behaviors of two previously studied one-parameter systems, namely the wedge and parabolic billiard. The billiard consists of a point mass moving in two dimensions under the influence of a constant gravitational field with a hyperbolic lower boundary. An iterative mapping between successive collisions with the lower boundary is derived analytically. The behavior of the system during transformation from the wedge to the parabola is investigated for a few specific cases. It is surprising that the nature of the transformation depends strongly on the parameter values. (c) 1999 American Institute of Physics.