Dynamical correlations of conserved quantities in the one-dimensional equal mass hard particle gas

We study a gas of point particles with hard-core repulsion in one dimension where the particles move freely in-between elastic collisions. We prepare the system with a uniform density on the infinite line. The velocities $\{v_i; i \in \mathbb{Z} \}$ of the particles are chosen independently from a thermal distribution. Using a mapping to the non-interacting gas, we analytically compute the equilibrium spatio-temporal correlations $\langle v_i^m(t) v_j^n(0)\rangle$ for arbitrary integers $m,n$. The analytical results are verified with microscopic simulations of the Hamiltonian dynamics. The correlation functions have ballistic scaling, as expected in an integrable model.