A Note on the Statistical Mechanics of Violent Relaxation of Phase‐Space Elements of Different Densities

Abstract

The statistical mechanics investigation of violent relaxation of phase-space elements of different densities first derived by Lynden-Bell is reexamined. It is found that the mass independence of the equations of motion of violent relaxation calls for a constraint on the volume of the phase-space elements used to formulate the statistical mechanics description of violent relaxation. In agreement with observations of astrophysical objects believed to have been subject to violent relaxation (e.g., clusters of galaxies), the coarse-grained phase-space distribution ${u{f}{7016}}$ --> of the final state in the nondegenerate limit turns into a superposition of Maxwellians, ${u{f}{7016}}$ --> -->i, of a common velocity dispersion. Thus, the velocity dispersion problem present in the investigation of Lynden-Bell is removed.