text: Systems that admit a treatment with the system size expansion may be described by a probability distribution , giving the probability of observing the system in state  at time .  may be, for example, a vector with elements corresponding to the number of molecules of different chemical species in a system. In a system of size  (intuitively interpreted as the volume), we will adopt the following nomenclature:  is a vector of macroscopic copy numbers,  is a vector of concentrations, and  is a vector of deterministic concentrations, as they would appear according to the rate equation in an infinite system.  and  are thus quantities subject to stochastic effects.
  A master equation describes the time evolution of this probability. Henceforth, a system of chemical reactions will be discussed to provide a concrete example, although the nomenclature of "species" and "reactions" is generalisable. A system involving  species and  reactions can be described with the master equation:
  Here,  is the system size,  is an operator which will be addressed later,  is the stoichiometric matrix for the system (in which element  gives the stoichiometric coefficient for species  in reaction ), and  is the rate of reaction  given a state  and system size .
   is a step operator, removing  from the th element of its argument. For example, . This formalism will be useful later.
  The above equation can be interpreted as follows. The initial sum on the RHS is over all reactions. For each reaction , the brackets immediately following the sum give two terms. The term with the simple coefficient −1 gives the probability flux away from a given state  due to reaction  changing the state. The term preceded by the product of step operators gives the probability flux due to reaction  changing a different state  into state . The product of step operators constructs this state .