  We study various temporal correlation functions of a tagged particle in
one-dimensional systems of interacting point particles evolving with
Hamiltonian dynamics. Initial conditions of the particles are chosen from the
canonical thermal distribution. The correlation functions are studied in finite
systems, and their forms examined at short and long times. Various
one-dimensional systems are studied. Results of numerical simulations for the
Fermi-Pasta-Ulam chain are qualitatively similar to results for the harmonic
chain, and agree unexpectedly well with a simple description in terms of
linearized equations for damped fluctuating sound waves. Simulation results for
the alternate mass hard particle gas reveal that - in contradiction to our
earlier results [1] with smaller system sizes - the diffusion constant slowly
converges to a constant value, in a manner consistent with mode coupling
theories. Our simulations also show that the behaviour of the Lennard-Jones gas
depends on its density. At low densities, it behaves like a hard-particle gas,
and at high densities like an anharmonic chain. In all the systems studied, the
tagged particle was found to show normal diffusion asymptotically, with
convergence times depending on the system under study. Finite size effects show
up at time scales larger than sound traversal times, their nature being
system-specific.
