Partition function in Statistical Physics

[Hymne]

Hi! I am for the moment reading a course in statistical physics where the author has definied not less then three diffrent partitionfunctions.

W, Z an Z which are called the microcanonical partitionfunction, canonical partitionfunction (?) and the grand canonical partitionfunction.

I have how ever a hard time keeping track of under which conditions to use which and what we assume that the system satifies when we use one of them.

Can somebody please help me se this subject clearer?!

 

[nonequilibrium]

microcanonical <-> E,V,N <-> isolated system (e.g. the universe (?), a very good coolbox)
canonical <-> T,V,N <-> closed rigid system that can exchange energy (e.g. a bottle in the coolbox)
grand canonincal <-> T,V,µ <-> open rigid system = exchange of energy and particles (e.g. the upper half of the bottle)

$$1/T := \frac{\partial S}{\partial <E>}$$ and $$-\mu / T := \frac{\partial S}{\partial <N>}$$

By their definitions you can see that they indeed characterize respective equilibrium in case of equality for both systems in contact.