Categories: Physics, Thermodynamic ensembles, Thermodynamics.

# Grand canonical ensemble

The **grand canonical ensemble** or **μVT ensemble**
extends the canonical ensemble
by allowing the exchange of both energy $U$ and particles $N$
with an external reservoir,
so that the conserved state functions are
the temperature $T$, the volume $V$, and the chemical potential $\mu$.

The derivation is practically identical to that of the canonical ensemble. We refer to the system of interest as $A$, and the reservoir as $B$. In total, $A\!+\!B$ has energy $U$ and population $N$.

Let $c_B(U_B)$ be the number of $B$-microstates with energy $U_B$. Then the probability that $A$ is in a specific microstate $s_A$ is as follows:

$\begin{aligned} p(s) = \frac{c_B\big(U - U_A(s_A), N - N_A(s_A)\big)}{\sum_{s_A} c_B\big(U \!-\! U_A(s_A), N \!-\! N_A(s_A)\big)} \end{aligned}$Then, as for the canonical ensemble,
we assume $U_B \gg U_A$ and $N_B \gg N_A$,
and approximate $\ln{p(s_A)}$
by Taylor-expanding $\ln{c_B}$ around $U_B = U$ and $N_B = N$.
The resulting probability distribution is known as the **Gibbs distribution**,
with $\beta \equiv 1/(kT)$:

Where the normalizing **grand partition function** $\mathcal{Z}(\mu, V, T)$ is defined as follows:

In contrast to the canonical ensemble,
whose thermodynamic potential
was the Helmholtz free energy $F$,
the grand canonical ensemble instead
minimizes the **grand potential** $\Omega$:

So $\mathcal{Z} = \exp(- \beta \Omega)$. This is proven in the same way as for $F$ in the canonical ensemble.

## References

- H. Gould, J. Tobochnik,
*Statistical and thermal physics*, 2nd edition, Princeton.