Long-time limit of correlation functions

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.

6 pages

References in corpus (4)

• Glass Transition in Confined Geometry
• Mode-coupling theory of the glass transition for confined fluids
• Completely monotone solutions of the mode-coupling theory for mixtures
• Mode-coupling theory for multiple decay channels

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• Scaling equations for mode-coupling theories with multiple decay channels
• Dynamics of fluids in quenched-random potential energy landscapes: a mode-coupling theory approach
• Mode-coupling theory of the glass transition for a liquid in a periodic potential