Long-time limit of correlation functions

TL;DR

This paper explores the conditions under which auto-correlation functions in equilibrium stochastic processes have a well-defined long-time limit, with implications for understanding glass transition phenomena.

Contribution

It provides general conditions linking spectral properties of correlation measures to the existence of long-time limits in dynamical theories.

Findings

Conditions for long-time limit existence are established.

Implications for mode-coupling theory of glass transition are discussed.

Non-trivial long-time limits indicate idealized glass states.

Abstract

Auto-correlation functions in an equilibrium stochastic process are well-characterized by Bochner's theorem as Fourier transforms of a finite symmetric Borel measure. The existence of a long-time limit of these correlation functions depends on the spectral properties of the measure. Here we provide conditions applicable to a wide-class of dynamical theories guaranteeing the existence of the long-time limit. We discuss the implications in the context of the mode-coupling theory of the glass transition where a non-trivial long-time limit signals an idealized glass state.