  This paper considers the problem of estimating the time auto-correlation
function for a quantity that is defined in configuration space, given a
knowledge of the mean-square displacement as function of time in configuration
space. The problem is particularly relevant for viscous liquids, where the
interesting time-scales are often beyond those reachable by computer
simulation. An approximate formula is derived which reduces the calculation of
the time auto- correlation function to a "double canonical" average. In this
approximation, in the case of Langevin dynamics the mean-square displacement
itself may be evaluated from the "double partition function" . The scheme
developed is illustrated by computer simulations of a simple one-dimensional
system at different temperatures, showing good agreement between the exact time
auto-correlation functions and those found by the approximation.
