Mixing and ‘‘violent relaxation’’ for the one-dimensional gravitational Coulomb gas

Ideas from the theory of dynamical systems are applied to a collection of N identical point masses interacting via one-dimensional ``gravity,'' with a two-body force F=-G sgn(xₐ-xb). This Hamiltonian system is viewed as a geodesic flow on a curved, but conformally flat, N-dimensional space, the curvature of which exhibits certain interesting features. In the large-N limit, the Ricci tensor can be written approximately in the form Rₐᵇ=-{Λ}δₐᵇ+Bₐᵇ, where {Λ} is a smooth, negative-definite quantity which, oftentimes, can be viewed as nearly constant, and Bₐᵇ is a positive sum of {δ} functions of codimension 1. The curvature K(u,{δ}x)==Rabcd{u}ᵇuᵈ{δ}xᵃ{δ}xᶜ associated with a small change {δ}xᵃ in the N-velocity uᵃ of the system is not negative definite. However, in the large-N limit, the smooth piece of K associated with a ``generic'' change {δ}xᵃ of a generic uᵃ is in fact negative, the fraction of uᵃ's and {δ}xᵃ's leading to a positive K decreasing at least as fast as N^-1. If the {δ}-function contributions to K could be ignored, as they can for D-dimensional ``gravity'' with D{≥}2, this would imply an average ``mixing'' very much akin to the astrophysicists' ``violent relaxation'' on a time scale {τ} given essentially as the free-fall time for the system. This conclusion fails, however, because of the singular contributions to K reflecting physical collisions which, for one-dimensional gravity, cannot be ignored. The similarities and differences between one- and three-dimensional ``gravitational'' systems are discussed, and it is argued that numerical simulations of one-dimensional systems need say little about real three-dimensional systems.