Gravity in one dimension: Diffusion in acceleration

Abstract

The one-dimensional gravitational system consists ofN parallel sheets of constant mass density. The sheets move perpendicular to their surface solely under their mutual gravitational attraction. When a pair has an encounter, they simply pass through each other. In this paper I consider the motion of a single sheet in an equilibrium ensemble. Under the assumption that the times separating encounters are random, I show that the acceleration and velocity(A, V) of a labeled sheet form a Markovian pair. Further, I prove that, in the limit of largeN, (1)the(A, V) process is deterministic, (2) the(A, V) process obeys Vlasov dynamics, and (3) that scaled fluctuations in(A, V) comprise a diffusion which obeys a generalized Ornstein-Uhlenbeck process with time-dependent drift and diffusion tensors.