Ensembles

An ensemble is just a collection: we imagine a collection of copies of the system, each in one of the allowed microstates. If the number of copies $\nu$ is much, much larger than $\Omega$, then each microstate will be represented with a frequency which reflects its probability: if $\nu_i$ is the number of copies in state $i$, we have $\nu_i=\nu p_i$. (We use $\nu$ for numbers of copies, and $n$ or $N$ for numbers of atoms. The former is hugely greater than the latter--it is just as well that it is only a theoretical concept.)

Then if we use $\lambda=1\ldots\nu$ to label the copies and $i$ to label the microstates,

$$\langle X\rangle={1\over\nu}\sum_\lambda X_\lambda ={1\over\nu}\sum_i \nu_i X_i=\sum_i p_i X_i$$

There are three kinds of ensembles commonly used in statistical physics. Where the real system is isolated, that is at fixed energy and particle number, the copies in the ensemble are also isolated from one another; this is called the microcanonical ensemble.

If the real system is in contact with a heat bath, that is at fixed temperature, the copies are assumed to be in thermal contact, with all the rest of the copies acting as a heat bath of any individual copy. This is called the canonical ensemble.

Finally, if the real system can exchange both heat and particles with a reservoir, at fixed temperature and chemical potential, the copies are also assumed to be in diffusive contact. This is called the grand canonical ensemble.