---
title: "Grand canonical ensemble"
sort_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.