Dynamics of supercooled liquids: Density fluctuations and mode coupling theory
We write equations of motion for density variables that are equivalent to Newton's equations. We then propose a set of trial equations parametrized by two unknown functions to describe the exact equations. These are chosen to best fit the exact Newtonian equations. Following established ideas, we choose to separate these trial functions into a set representing integrable motions of density waves, and a set containing all effects of non-integrability. The density waves are found to have the dispersion of sound waves, and this ensures that the interactions between the independent waves are minimized. Furthermore, it transpires that the static structure factor is fixed by this minimum condition to be the solution of the Yvon-Born-Green equation. The residual interactions between density waves are explicitly isolated in their Newtonian representation and expanded by choosing the dominant objects in the phase space of the system, that can be represented by a dissipative term with memory and a random noise. This provides a mapping between deterministic and stochastic dynamics. Imposing the fluctuation-dissipation theorem allows us to calculate the memory kernel. We write exactly the expression for it, following two different routes, i.e. using explicitly Newton's equations, or instead, their implicit form, that must be projected onto density pairs, as in the development of the well established mode coupling theory. We compare these two ways of proceeding, showing the necessity to enforce a new equation of constraint for the two schemes to be consistent. Thus, while in the first 'Newtonian' representation a simple Gaussian approximation for the random process leads easily to the mean spherical approximation for the statics and to MCT for the dynamics of the system, in the second case higher levels of approximation are required to have a fully consistent theory.