In section 13.3.2 of Statistical Mechanics: Theory and Molecular Simulation by Mark E. Tuckerman, the author derives the Green-Kubo relations for the diffusion constant. In the derivation, he makes the following claim:
Recall that in equilibrium, the velocity (momentum) distribution is a product of independent Gaussian distributions. Hence, $\langle \dot{x}_i\dot{x}_j \rangle$ is $0$, and moreover, all cross correlations $\langle \dot{x}_i (0) \dot{x}_j (t) \rangle$ vanish when $i\neq j$.
However, I have doubts regarding the second claim ($\langle \dot{x}_i (0) \dot{x}_j (t) \rangle = 0$ when $i\neq j$). I am able to prove it for non-interacting particles using the Langevin equation (with the assumption that the white noise terms corresponding to different particles are uncorrelated). However, I am not sure that this statement is true for a system of interacting particles. Is there a rigorous proof that the velocity cross-correlations vanish for different particles even in the presence of interactions?