  We study a non-ergodic transition in a many-body Langevin system. We first
derive an equation for the two-point time correlation function of density
fluctuations, ignoring the contributions of the third- and fourth-order
cumulants. For this equation, with the average density fixed, we find that
there is a critical temperature at which the qualitative nature of the
trajectories around the trivial solution changes. Using a method of dynamical
system reduction around the critical temperature, we simplify the equation for
the time correlation function into a two-dimensional ordinary differential
equation. Analyzing this differential equation, we demonstrate that a
non-ergodic transition occurs at some temperature slightly higher than the
critical temperature.
