Abstract
We present a simple, linear, partial-differential equation for the
density-density correlation function in a glass-forming system. The
equation is written down on the basis of fundamental and general
considerations of linearity, symmetry, stability, thermodynamic
irreversibility and consistency with the equation of continuity, that
is, conservation of matter. The dynamical properties of the
solutions show a change in behaviour characteristic of the
liquid-glass transition as a function of one of the parameters
(temperature). The equation can be shown to lead to the simplest
mode-coupling theory of glasses and provides a partial justification
of this simplest theory. It provides also a method for calculating
the space-dependence of the correlation functions not available
otherwise.
The results suggest certain differences in behaviour between
glassy solids and glass-forming liquids that may be accessible to
experiment. A brief discussion is presented of how the method can be
applied to other systems such as sand-piles and vortex-glasses in
Type II superconductors.
The methods discussed above can and have been used to construct an
undergraduate research project which provides the student with
experience in a wide variety of numerical and analytic techniques.