We refine the derivation of the Boltzmann equation by considering that the molecules passing through the interfaces of a volume element of physical space and velocity space exhibit different velocity distribution functions and number densities. The resulting equation has a time parameter close to the relaxation time and degenerates into the conventional Boltzmann equation when this parameter takes a value of zero. By considering the macroscopic averaging of mass, momentum, and energy, the corresponding continuity, momentum, and energy equations are obtained. Compared with the extended Navier–Stokes equations, the momentum and energy equations contain additional terms to represent the external forces.

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