From 197bbd585d86ca7091d13144b89441f64e9cfc6a Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 11 Jul 2021 15:29:51 +0200
Subject: Expand knowledge base

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+---
+title: "Grand canonical ensemble"
+firstLetter: "G"
+publishDate: 2021-07-11
+categories:
+- Physics
+- Thermodynamics
+- Thermodynamic ensembles
+
+date: 2021-07-08T11:01:11+02:00
+draft: false
+markup: pandoc
+---
+
+# Grand canonical ensemble
+
+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** $\Omega$:
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            \Omega(T, V, \mu)
+            &\equiv - k T \ln{\mathcal{Z}}
+            \\
+            &= \expval{U_A} - T S_A - \mu \expval{N_A}
+        \end{aligned}
+    }
+\end{aligned}$$
+
+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.
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