## Physics Friday 84

The canonical ensemble is a powerful tool for statistical mechanics. Let us now illustrate another powerful formalism, an extension known as the “grand canonical” formalism.

In the canonical formalism, our system is in contact with a thermal reservoir with which it may exchange energy; the reservir is large enough that it’s temperature be treated as a constant (thus giving it an entropy which varies linearly with energy). In the grand canonical formalism, we add the ability to exchange particles, as well as energy, with the particle reservoir large enough that the chemical potential  is treated as a constant.

If we treat the combination of our system and the reservoir as a closed system with every microstate equally probable, we have for a state where our system has energy Ei and particle number Ni, the fractional occupation is the number of microstates of the reservoir for which its energy is EtEi and it’s particle number is NtNi divided by the total number of microstates for the system and reservoir combination with total energy Et and a total of Nt particles:

However, using the entropy definition , we can rewrite this in terms of the reservoir and total entropies (as functions of energy and particle number):
.
As with our canonical formalism, we can expand the entropy as a Taylor series in energy about EtU and in particle number about the average particle number N
, (using ).
And via additivity of the entropy, .
Thus
,
where  is the thermodynamic beta and  is the “grand potential”. As with the canonical ensemble, we normalize this: , where
 is the “grand canonical partition function”, and the sum is over all states, with state s having energy Es and Ns particles.

We see that just as , we can solve for the grand potential:
.
Let us define . Now, the expectation value for the number of particles is
,
or, using the chain rule,
.
Similarly, one can show that
,
the variance in the particle number.

Now, consider the logarithmic partial derivative of Ƶ with respect to β:
,
and we have our internal energy. We can also show by a second differentiation that
.

From our equation for the grand potential, we see
.

Next, consider the fundamental thermodynamic relation . Soving for the mechanical work term,
. Now, if we continue to hold β (and thus T) and μ constant, the right hand side of the previous relation is just –:
.
From this, one can obtain the equation of state
.