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.

Dynamics of fluids in quenched-random potential energy landscapes: a mode-coupling theory approach

“…obeyed by the infinite-time limits f c q = lim t→∞ φ c q (t) and f s q = lim t→∞ φ s q (t), 37,38 whose existence has been proved recently. 93 The results are shown in Fig. 4.…”

Rapid onset of molecular friction in liquids bridging between the atomistic and hydrodynamic pictures

“…In equilibrium, only positive frequencies are needed as ReĈðωÞ ¼ ReĈðÀωÞ, and the integrals are real-valued. Next, we introduce a memory function of C(t) solely by invoking results from complex analysis 68 .ĈðzÞ as above is a holomorphic function with ReĈðzÞ ≥ 0, i.e., iĈðzÞ is of Herglotz-Nevanlinna type, and ðIm zÞ jĈðzÞj is bounded in C þ . Suppose that C(t) has a regular short-time expansion, Cðt !…”

Complex dynamics induced by strong confinement – From tracer diffusion in strongly heterogeneous media to glassy relaxation of dense fluids in narrow slits

“…To describe dense liquids in such confinement the MCT has been extended [44,114] relying on symmetry-adapted modes that account for the broken translational symmetry perpendicular to the walls. A striking prediction of the MCT in slit geometry has been the emergence of a multiple reentrant glass transition in the nonequilibrium state diagram as a function of the slit width along lines of constant packing fraction [44,115,116]. This multiple reentrance is attributed to a complex competition between the layering induced by the walls and the local caging.…”

“…A natural question that arises is if these equations still provide unique solutions that represent correlation functions as has been proven in bulk systems [1,[121][122][123] and if the long-time limits can be shown to exist [116]. The first question has been answered positively recently for general multiple relaxation channels [124] which arise for example also for molecular systems [125,126].…”