Abstract
AbstractFor almost thirty years, mode-coupling theory has been the most widely discussed and used but also the most controversial theory of the glass transition. In this paper we briefly review the reasons for both its popularity and its controversy. We emphasize the need for the development of approaches that would be able to evaluate corrections to and extensions of the existing (standard) mode-coupling theory. Next, we review our diagrammatic formulation of the dynamics of interacting Brownian particles.We show that within this approach the standardmode-coupling theory can be derived in a very simple way. Finally, we use our diagrammatic approach to calculate two corrections to the mode-coupling theory's expression for the so-called irreducible memory function. These corrections involve re-summations of well-defined classes of non-mode-coupling diagrams.
Highlights
Since the publication, almost thirty years ago, of three nearly coincidental papers by Leutheusser [1], Begtzelius, Götze and Sjölander [2], and Das, Mazenko, Ramaswamy and Toner [3], mode-coupling theory has been the most widely used and discussed, and the most controversial theoretical approach to the glass transition problem
We should emphasize that our diagrammatic expansion was not derived from a fieldtheoretical approach
To write down Eq (45) in a slightly more compact form we used the function v defined in Eq (12) and we introduced the mode-coupling theory’s memory matrix that has the delta function originating from translational invariance factored out, MMCT(k, k1; t) = MMCT(k; t)(2π)3δ(k − k1), (46)
Summary
Almost thirty years ago, of three nearly coincidental papers by Leutheusser [1], Begtzelius, Götze and Sjölander [2], and Das, Mazenko, Ramaswamy and Toner [3], mode-coupling theory has been the most widely used and discussed, and the most controversial theoretical approach to the glass transition problem. Critics of the theory emphasize the fact that it can only describe the initial three decades of the slowing down and that it predicts a spurious (non-existent) ergodicity breaking transition They point out the discrepancy between the mode-coupling temperature or volume fraction obtained from fitting the simulation results and the corresponding quantities predicted by the theory. We shall mention here the so-called generalized mode-coupling approach This line of research was started when we recognized [30] that by moving mode-coupling theory’s factorization approximation to a higher level correlation function the location of the ergodicity breaking transition predicted by the theory can be moved towards the empirical transition determined by fitting simulational data to mode-coupling-like power laws. We will evaluate the simplest corrections to the mode-coupling theory’s expression for the so-called irreducible memory function
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