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

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.