Velocity-Field Theory, Boltzmann's Transport Equation and Geometry

TL;DR

This paper develops a novel field theory formulation of Boltzmann's equation focusing on velocity fields, explicitly deriving collision terms and connecting geometric and path-integral methods to define distribution functions.

Contribution

It introduces a velocity-field based field theory approach to Boltzmann's equation, explicitly derives collision terms for four-body interactions, and links geometric and path-integral techniques to distribution functions.

Findings

Explicit collision term for four-body interaction derived

Fluctuation distinguished from quantum effects

Connection between geometry, path-integral, and distribution functions established

Abstract

Boltzmann equation describes the time development of the velocity distribution in the continuum fluid matter. We formulate the equation using the field theory where the {\it velocity-field} plays the central role. The matter (constituent particles) fields appear as the density and the viscosity. {\it Fluctuation} is examined, and is clearly discriminated from the quantum effect. The time variable is {\it emergently} introduced through the computational process step. The collision term, for the (velocity)**4 potential (4-body interaction), is explicitly obtained and the (statistical) fluctuation is closely explained. The present field theory model does {\it not} conserve energy and is an open-system model. (One dimensional) Navier-Stokes equation or Burger's equation, appears. In the latter part, we present a way to directly define the distribution function by use of the geometry, appearing in the mechanical dynamics, and Feynman's path-integral.