BOGOLIUBOV-BORN-GREEN-KIRKWOOD-YVON HIERARCHY FOR HIGHER FORMS

Jolicoeur and Le Guillou have recently proposed [Phys. Rev. A 40, 5815 (1989)] a very interesting derivation of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy using functional techniques. In this paper we generalized their derivation to higher-form distributions in phase space. At the heart of our method there is a path integral for classical mechanics that we put forward some time ago. This path integral describes the dynamics not only of scalar distributions in phase space but also of p-form valued densities [E. Gozzi, M. Reuter, and W. D. Thacker, Phys. Rev. D 40, 3363 (1989)]. The distribution functions entering this infinite set of coupled integro-differential equations (BBGKY hierarchy) carry a double grading now: besides the level in the hierarchy, they are characterized by their degree as differential forms. The higher forms are related to the dynamics of the Jacobi fields and therefore contain information about the behavior of nearby trajectories. This kind of information could be used in the study of turbulence, for instance.