Is Boltzmann’s Equation Physically Insufficient?

Abstract

Boltzmann’s equation describes the dynamics of the one particle phase space distribution density including two particle interactions. These drive the system towards equilibrium. The physical properties of this equilibrium turn out to be those of an ideal gas, giving the leading order of the equation of state, the pressure, etc, only. No viral corrections originating from the two particle interactions appear, as one finds them in equilibrium statistical mechanics. This means the Boltzmann dynamics is insufficient to imply the proper equilibrium. It has to be upgraded, clearly, in its so called flow terms, which contribute only in order n (particle density) corresponding to a free motion between collisions. If properly derived, the equation of motion contains also order n contributions in the flow terms, weighted by the real part of the forward scattering amplitude. Then the equilibrium limit of the dynamics coincides with the findings of equilibrium statistical physics. Various implications can be identified. It is pointed out, that the corresponding upgrade has to be checked carefully in interdisciplinary modeling, as is used, e.g., in traffic flow or granular systems descriptions.