Non-ergodic transitions in many-body Langevin systems: a method of dynamical system reduction

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 thirdand fourth-order cummulants. 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 twodimensional ordinary differential equation. Analyzing this differential equation, we demonstrate that a non-ergodic transition occurs at some temperature slightly higher than the critical temperature. Non-ergodic transitions in many-body Langevin systems 2