Dynamics of supercooled liquids: densityfluctuations and mode coupling theory

Abstract

We write equations of motion for density variables that areequivalent to Newton's equations. We then propose a set of trialequations parametrized by two unknown functions to describe theexact equations. These are chosen to best fit the exactNewtonian equations. Following established ideas, we choose toseparate these trial functions into a set representingintegrable motions of density waves, and a set containing alleffects of non-integrability. The density waves are found tohave the dispersion of sound waves, and this ensures that theinteractions between the independent waves are minimized.Furthermore, it transpires that the static structure factor isfixed by this minimum condition to be the solution of theYvon-Born-Green equation. The residual interactionsbetween density waves are explicitly isolated in their Newtonianrepresentation and expanded by choosing the dominant objects inthe phase space of the system, that can be represented by adissipative term with memory and a random noise. This provides amapping between deterministic and stochastic dynamics. Imposingthe fluctuation-dissipation theorem allows us to calculatethe memory kernel. We write exactly the expression for it,following two different routes, i.e. using explicitly Newton'sequations, or instead, their implicit form, that must beprojected onto density pairs, as in the development of thewell established mode coupling theory. We compare thesetwo ways of proceeding, showing the necessity to enforce a newequation of constraint for the two schemes to be consistent. Thus, while in the first `Newtonian' representation a simpleGaussian approximation for the random process leads easily tothe mean spherical approximation for the statics and toMCT for the dynamics of the system, in the second case higherlevels of approximation are required to have a fully consistenttheory.