Particle swarms in gases: the velocity-average evolution equations from Newton's law

The evolution equation for a generic average quantity relevant to a swarm of particles homogeneously dispersed in a uniform gas, is obtained directly from the Newton's law, without having recourse to the (intermediary) Boltzmann equation. The procedure makes use of appropriate averages of the term resulting from the impulsive force (due to collisions) in the Newton's law. When the background gas is assumed to be in thermal equilibrium, the obtained evolution equation is shown to agree with the corresponding one following from the Boltzmann equation. But the new procedure also allows to treat physical situations in which the Boltzmann equation is not valid, as it happens when some correlation exists (or is assumed) between the velocities of swarm and gas particles.