is the gradient operator, is the dot product, is a shorthand for the momentum analogue of , and , , are Cartesian unit vectors. Final statement Dividing () by and substituting into () gives: In this context, is the force field acting on the particles in the fluid, and is the mass of the particles. The term on the right hand side is added to describe the effect of collisions between particles; if it is zero then the particles do not collide. The collisionless Boltzmann equation, where individual collisions are replaced with long-range aggregated interactions, e.g. Coulomb interactions, is often called the Vlasov equation. This equation is more useful than the principal one above, yet still incomplete, since cannot be solved unless the collision term in is known. This term cannot be found as easily or generally as the others – it is a statistical term representing the particle collisions, and requires knowledge of the statistics the particles obey, like the Maxwell–Boltzmann, Fermi–Dirac or Bose–Einstein distributions. The collision term (Stosszahlansatz) and molecular chaos Two-body collision term A key insight applied by Boltzmann was to determine the collision term resulting solely from two-body collisions between particles that are assumed to be uncorrelated prior to the collision. This assumption was referred to by Boltzmann as the "" and is also known as the "molecular chaos assumption". Under this assumption the collision term can be written as a momentum-space integral over the product of one-particle distribution functions: where and are the momenta of any two particles (labeled as A and B for convenience) before a collision, and are the momenta after the collision, is the magnitude of the relative momenta (see relative velocity for more on this concept), and is the differential cross section of the collision, in which the relative momenta of the colliding particles turns through an angle into the element of the solid angle , due to the collision. Simplifications to the collision term Since much of the challenge in solving the Boltzmann equation originates with the complex collision term, attempts have been made to "model" and simplify the collision term. The best known model equation is due to Bhatnagar, Gross and Krook. The assumption in the BGK approximation is that the effect of molecular collisions is to force a non-equilibrium distribution function at a point in physical space back to a Maxwellian equilibrium distribution function and that the rate