---
title: "Grand canonical ensemble"
date: 2021-07-11
categories:
- Physics
- Thermodynamics
- Thermodynamic ensembles
layout: "concept"
---

The **grand canonical ensemble** or **μVT ensemble**
extends the [canonical ensemble](/know/concept/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 $\beta \equiv 1/(kT)$:

$$\begin{aligned}
    \boxed{
        p(s_A) = \frac{1}{\mathcal{Z}} \exp\!\Big(\!-\! \beta \: \big( U_A(s_A) \!-\! \mu N_A(s_A) \big) \Big)
    }
\end{aligned}$$

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

$$\begin{aligned}
    \boxed{
        \mathcal{Z} \equiv \sum_{s_A}^{} \exp\!\Big(\!-\! \beta \: \big( U_A(s_A) - \mu N_A(s_A) \big) \Big)
    }
\end{aligned}$$

In contrast to the canonical ensemble,
whose [thermodynamic potential](/know/concept/thermodynamic-potential/)
was the Helmholtz free energy $F$,
the grand canonical ensemble instead
minimizes the **grand potential** $\Omega$:

$$\begin{aligned}
    \boxed{
        \Omega(T, V, \mu)
        \equiv - k T \ln{\mathcal{Z}}
        = \Expval{U_A} - T S_A - \mu \Expval{N_A}
    }
\end{aligned}$$

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



## References
1.  H. Gould, J. Tobochnik,
    *Statistical and thermal physics*, 2nd edition,
    Princeton.