Difference between canonical ensemble and grand canonical ensemble
Answers
The canonical ensemble applies to a system coupled to a large environment with which it can exchange energy (i.e., an energy “reservoir”). Suppose that we can describe the whole system + reservoir by a microcanonical ensemble. Then, the probability of the system being in a particular configuration of energy E is simply proportional to the number of ways of depositing the remaining energy U-E in the reservoir. The probabilities implied by this procedure define the canonical ensemble. This construction also motivates the definition of temperature as the derivative of the number of configurations available to the reservoir as a function of its energy content. Note that the description of the canonical ensemble is very appropriate to real world materials, for instance, that absorb/emit radiation to/from their environment.
The grand canonical ensemble applies to a system coupled to a large environment with which it can exchange energy and particles (i.e., a particle and energy reservoir). The construction of this ensemble is completely analogous to the canonical ensemble: we apply the microcanonical ensemble to the system + reservoir and then consider the probability of a particular configuration of the system. This construction motivates the definition of the chemical potential.