If the number of parts in the system is allowed to vary among the systems in the ensemble (as in a grand ensemble where the number of particles is a random quantity), then it is a probability distribution over an extended phase space that includes further variables such as particle numbers (first kind of particle), (second kind of particle), and so on up to (the last kind of particle; is how many different kinds of particles there are). The ensemble is then represented by a joint probability density function . The number of coordinates varies with the numbers of particles.