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Boltzmann Equation Field Theory I: Ensemble Averages

Jun Yan Lau

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.

Boltzmann Equation Field Theory I: Ensemble Averages

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.
Paper Structure (23 sections, 96 equations)

This paper contains 23 sections, 96 equations.

Table of Contents

  1. Introduction
  2. Mathematical Methods
  3. Poisson Sampling
  4. The Law of Large Numbers
  5. From Liouville's to Boltzmann's
  6. Self-consistent Hamiltonian
  7. Functional Differentiation
  8. Functional Integration
  9. Assigning distributions to particles
  10. Entropy Maximisation Criterion
  11. Ensemble Averages
  12. The Maxwellian
  13. Background Correlation Functions
  14. Discussion
  15. The Probability Volume of the Macrostate, $\Omega_N$
  16. ...and 8 more sections