Field Theory Approach to Classical $N$-Particle Systems In and Out of Equilibrium

We present an approach to solving the evolution of a classical $N$-particle ensemble based on the path integral approach to classical mechanics. This formulation provides a perturbative solution to the Liouville equation in terms of a propagator which can be expanded in a Dyson series. We show that this perturbative expansion exactly corresponds to an iterative solution of the BBGKY-hierarchy in orders of the interaction potential. Using the path integral formulation, we perform a Hubbard-Stratonovich transformation (HST) to obtain an effective field theoretic description in terms of macroscopic fields, which contains the full microscopic dynamics of the system in its vertices. Naturally, the HST leads to a new perturbative expansion scheme which contains an infinite order of microscopic interactions already at the lowest order of the perturbative expansion. Our approach can be applied to in and out of equilibrium systems with arbitrary interaction potentials and initial conditions. We show how (unequal-time) cumulants of the Klimontovich phase space densities can be computed within this framework and derive results for density and momentum correlations for a spatially homogeneous system. Under the explicit assumptions for the interaction potential and initial conditions, we show that well-known results related to plasma oscillations and the Jeans instability criterion for gravitational collapse can be recovered in the lowest order perturbative expansion and that both are the effect of the same collective behaviour of the many-body system.