text: Consider particles described by , each experiencing an external force  not due to other particles (see the collision term for the latter treatment).
  Suppose at time  some number of particles all have position  within element  and momentum  within . If a force  instantly acts on each particle, then at time  their position will be  and momentum . Then, in the absence of collisions,  must satisfy
  Note that we have used the fact that the phase space volume element  is constant, which can be shown using Hamiltons equations (see the discussion under Liouvilles theorem). However, since collisions do occur, the particle density in the phase-space volume  changes, so
  where  is the total change in . Dividing () by  and taking the limits  and , we have
  The total differential of  is:
  where  is the gradient operator,  is the dot product,
  is a shorthand for the momentum analogue of , and , ,  are Cartesian unit vectors.