Long-time limit of correlation functions (Q2878625)

Corresponding to ergodic behaviour, the correlation functions are supposed to approach zero at large times. For non-ergodic behaviour, the question is thus if correlation functions display a non-trivial long-time limit. The aim of this paper is to find general conditions applicable to a wide class of dynamical theories such that the long-time limit of the correlation function is guaranteed to exist. To do this, the paper starts recalling basic properties of time correlation functions and continues with the formulation of the problem, model equations and the study of spectral properties, obtaining the existence of long-time limits. As an application, implications for mode-coupling theory are analyzed and the existence of long-time limits for Newtonian dynamics is proved, which provides the key to characterize idealized glass states and the associated non-equilibrium state diagram.