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

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 properties of the fluid matter (fluid particles) appear as the density and the viscosity. {\it Statistical fluctuation} is examined, and is clearly discriminated from the quantum effect. The time variable is {\it emergently} introduced through the computational process step. Besides the ordinary potential, the general velocity potential is introduced. The collision term, for the Higgs-type velocity potential, is explicitly obtained and the (statistical) fluctuation is closely explained. The system is generally {\it non-equilibrium}. The present field theory model does {\it not} conserve energy and is an open-system model. One dimensional Navier-Stokes equation, i.e., Burgers equation, appears. In the latter part of the text, we present a way to directly define the distribution function by use of the geometry, appearing in the energy expression, and Feynman's path-integral.