Since the probability of N molecules, which all have r and p within d 3 r d 3 p , is in question, at the heart of the equation is a quantity f which gives this probability per unit phase-space volume, or probability per unit length cubed per unit momentum cubed, at an instant of time t. This is a probability density function : f (r, p, t), defined so that, d N = f ( r , p , t ) d 3 r d 3 p is the number of molecules which all have positions lying within a volume element d 3 r about r and momenta lying within a momentum space element d 3 p about p, at time t. Integrating over a region of position space and momentum space gives the total number of particles which have positions and momenta in that region: