Mode-coupling theory

In this set of lecture notes we review the mode-coupling theory of the glass transition from several perspectives. First, we derive mode-coupling equations for the description of density fluctuations from microscopic considerations with the use the Mori-Zwanzig projection operator technique. We also derive schematic mode-coupling equations of a similar form from a field-theoretic perspective. We review the successes and failures of modecoupling theory, and discuss recent advances in the applications of the theory. 1 Important Phenomenology for MCT Since our objective will be to sketch a derivation of what we will call ModeCoupling Theory (MCT), we will focus our attention on one observable in particular, namely density fluctuations. For this we will first define some of the concepts needed to do so. We want to calculate a specific time correlation function. In general, such a function is expressed as follows, C(t) = 〈A(t)A(0)〉. (1) It is an ensemble average of the evolution of the fluctuations of a variable in time, at equilibrium. As seen in Fig. 1, A(t) fluctuates around its average value in equilibrium, while C(t) measures the correlation of A at one time with the value of A at another time. The density or particles in a liquid can be one example of A(t),