Boltzmann Equation Field Theory I: Ensemble Averages

I present an unbiased method of mapping particles to distribution functions and vice versa. This method alone defines the canonical formulation of statistical mechanics, since it can be used to derive the principle of maximum entropy in both Boltzmann's paradigm and Gibbs' paradigm. A rigorous definition of the macrostate enables application of this statistical mechanical theory to self-gravitating systems, by decoupling time-averages and ensemble averages. I compute two-point correlation functions for self-gravitating and electrostatic systems.