  We consider a one-dimensional gas of hard point particles in a finite box
that are in thermal equilibrium and evolving under Hamiltonian dynamics. Tagged
particle correlation functions of the middle particle are studied. For the
special case where all particles have the same mass, we obtain analytic results
for the velocity auto-correlation function in the short time diffusive regime
and the long time approach to the saturation value when finite-size effects
become relevant. In the case where the masses are unequal, numerical
simulations indicate sub-diffusive behaviour with mean square displacement of
the tagged particle growing as t/ln(t) with time t. Also various correlation
functions, involving the velocity and position of the tagged particle, show
damped oscillations at long times that are absent for the equal mass case.
