The Boltzmann equation for a fluid with a body force is
where is the number of molecules at time t with velocities in the range and position in the range . The difference between the Boltzmann equation when there is no body force present and when there is a body force is an extra term: . In the lattice Boltzmann equation we are looking to add a similar term to incorporate a body force. Since, however, the velocity of all the `particles' is constant in the lattice Boltzmann model we can not simply introduce a term with exactly the same form but must instead introduce a term which will modify the fluid momentum.