Boltzmann Equation Field Theory I: Ensemble Averages

TL;DR

This paper introduces an unbiased method for mapping particles to distribution functions, establishing a canonical statistical mechanics framework applicable to self-gravitating systems and deriving correlation functions.

Contribution

It presents a novel, unbiased approach to connect particles and distribution functions, foundational for statistical mechanics and applicable to complex systems like self-gravity.

Findings

Derived two-point correlation functions for self-gravitating systems

Established a rigorous macrostate definition for statistical mechanics

Linked maximum entropy principles to particle-distribution mappings

Abstract

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.