## some physics, why not

Mostly for my benefit, I’m looking for a simple way to compare the canonical and grand canonical ensembles in statistical physics. How about this…

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The canonical ensemble is a load of “systems”, each of which has a set of energy levels, and can exchange energy with all the other systems.

The probability of a system being in the ith state, whose energy is E_i, is:

$P_i = \exp(-E_i / kT) / Z$

where Z is the system’s partition function.

The systems don’t have to be identical, but different systems have different Z’s.

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The grand canonical ensemble is a load of “systems”, each of which has a set of “legal” particle numbers: $N_1, N_2, N_3$, … (probably 0, 1, 2…) and has a set of energy levels for each particle number, and can exchange energy and particles with all the other systems.

The probability of a system being in state (r,s), meaning it has $N_s$ particles and total energy $E_r$, is:

$P_{r,s} = exp(-E_r - \mu N_s) / X$

where X is the system’s grand partition function. $\mu$ is called chemical potential. Now to explain that…