On the Hamiltonian structure of the BBGKY hierarchy

It is now well known that the Vlasov-Poisson system is a noncanonical Hamiltonian field theory [P. Morrison, Phys. Lett. A 80, 383, (1980)]; however, it is less well known that there exists families of Hamiltonian closures , Vlasov-like Hamiltonian field theories for the coupled dynamics of 1-point and 2-point functions, coupled 1-point, 2-point, and 3-point functions, and so on up the chain [J. Marsden, P. Morrison, & A. Weinstein, Cont. Math. 28, 115 (1984)]. We will review this structure and analyze the system for the coupled 1-point and 2-point functions, a system that can produce the Lenard-Balescu collision operator following a natural procedure.