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Complex Dynamics of Glass-Forming LiquidsA Mode-Coupling Theory$
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Wolfgang Götze

Print publication date: 2008

Print ISBN-13: 9780199235346

Published to Oxford Scholarship Online: May 2009

DOI: 10.1093/acprof:oso/9780199235346.001.0001

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Foundations of the Mode-Coupling Theory for the Evolution of Glassy Dynamics in Liquids

Foundations of the Mode-Coupling Theory for the Evolution of Glassy Dynamics in Liquids

Chapter:
(p.177) 4 Foundations of the Mode-Coupling Theory for the Evolution of Glassy Dynamics in Liquids
Source:
Complex Dynamics of Glass-Forming Liquids
Author(s):

Wolfgang Götze

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199235346.003.0004

In this chapter, mode-coupling equations of motion for a correlation-function description of the dynamics of simple liquids and colloids are derived and their mathematical properties are analysed. The central concepts are expressions for the fluctuating-force kernels as mode-coupling polynomials of the density-fluctuation-correlation functions. The arrested parts of the latter are solutions of a fixed-point equation, which exhibits bifurcation singularities. The simplest ones are generic and degenerate fold bifurcations, which describe liquid–glass transitions. Schematic models are introduced in order to exemplify by elementary calculations different scenarios for the correlation arrest. The transitions in hard-sphere systems and in square-well systems are explained. Evolutions of the dynamics due to the approach of control parameters towards the critical values for the bifurcation points are analysed in order to show that the theoretical results are similar to those observed for the glassy dynamics of liquids.

Keywords:   mode-coupling equations of motion, mode-coupling polynomials, fixed point equation, bifurcation singularities, degenerate fold bifurcations, generic fold bifurcations, hard-sphere systems, square-well systems, critical values

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