Abstract:

A typical liquid can be cooled down so as to avoid crystallisation resulting in what is called a supercooled liquid. Its static properties are mere continuations of those of a liquid. However several new phenomena emerges in dynamics. The most prominent of them is the tremendous slowing down of the dynamics, referred to as a glass transition. The global relaxation time grows from a pi- cosecond scale at the crystallisation transition up to a macroscopic value for low temperatures. Understanding of the phenomena in the low temperature state of supercooled liquids, in particular the tremendous slowing down, is one of the longstanding open problems of the out of equilibrium statistical physics. Different scenarios were proposed to explain the observed low-T behaviour of supercooled liquids. In this thesis we follow the one based on the results of gen- eralised spin glass models (mean-field in nature) and the Mode-Coupling Theory (MCT). This scenario reproduces well the initial slow down of the dynamics but predicts a spurious transition to non-ergodic phase for low temperatures. Thus a problem naturally arises: how to enhance the model so as to get a correct behaviour at low temperatures and to eliminate the transition ? The main aim of the thesis is to construct such a generalisation. The general expectation is that corrections to MCT modify the low-T theory. However the main drawback of the MCT-based scenario is the uncontrolled nature of the approximation lying at the heart of MCT what makes the computation of cor- rections within the original projector operator formalism an almost unsolvable problem. A promising way to circumvent this problem is a rederivation of MCT within a field theoretical context. Revision of previous attempts reveals their inconsistency caused by the violation of the time-reversal symmetry (TRS) in perturbation series. Resolving this issue results in a correct derivation of MCT within a field theoretical context and yields perturbation series respecting TRS. This provides a direct way to test a structural stability of MCT and, thus, provides an insight on the very important problem of whether the transition cutoff can be understood within some refined approximation or has more fun- damental foundations. The result is that MCT is only a mean-field theory of the glass transition, so that perturbative corrections are irrelevant. Within this context we also explore the mapping via a Cole-Hopf transformation between the supercooled liquids and reaction-diffusion systems. The above mentioned results apply to the supercooled liquids above the transition i.e. to equilibrium dynamics. The last chapter deals with their partial generalisation to low temperatures where the system never reaches equilibrium during the observation and the global relaxation time is, effectively, infinite. Analysis turns out to be more complicated in that case since a bunch of new phenomena like ageing of physical properties comes into play.

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