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

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.