volume, or probability per unit length cubed per unit momentum cubed, at an instant of time . This is a probability density function: , defined so that, is the number of molecules which all have positions lying within a volume element about and momenta lying within a momentum space element about , at time . Integrating over a region of position space and momentum space gives the total number of particles which have positions and momenta in that region: which is a 6-fold integral. While is associated with a number of particles, the phase space is for one-particle (not all of them, which is usually the case with deterministic many-body systems), since only one and is in question. It is not part of the analysis to use , for particle 1, , for particle 2, etc. up to , for particle N. It is assumed the particles in the system are identical (so each has an identical mass ). For a mixture of more than one chemical species, one distribution is needed for each, see below. Principal statement The general equation can then be written as where the "force" term corresponds to the forces exerted on the particles by an external influence (not by the particles themselves), the "diff" term represents the diffusion of particles, and "coll" is the collision term – accounting for the forces acting between particles in collisions. Expressions for each term on the right side are provided below. Note that some authors use the particle velocity instead of momentum ; they are related in the definition of momentum by . The force and diffusion terms 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 Hamilton's equations (see the discussion under Liouville's 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