From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 20 Oct 2022 18:25:31 +0200
Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

---
 .../know/concept/grand-canonical-ensemble/index.md | 32 +++++++++++-----------
 1 file changed, 16 insertions(+), 16 deletions(-)

(limited to 'source/know/concept/grand-canonical-ensemble')

diff --git a/source/know/concept/grand-canonical-ensemble/index.md b/source/know/concept/grand-canonical-ensemble/index.md
index 4f66fd9..62ca896 100644
--- a/source/know/concept/grand-canonical-ensemble/index.md
+++ b/source/know/concept/grand-canonical-ensemble/index.md
@@ -11,18 +11,18 @@ 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 $N$
+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 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 $N$.
+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:
+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:
 
 $$\begin{aligned}
     p(s)
@@ -30,11 +30,11 @@ $$\begin{aligned}
 \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 $N_B = N$.
+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 $\beta \equiv 1/(kT)$:
+with $$\beta \equiv 1/(kT)$$:
 
 $$\begin{aligned}
     \boxed{
@@ -42,7 +42,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where the normalizing **grand partition function** $\mathcal{Z}(\mu, V, T)$ is defined as follows:
+Where the normalizing **grand partition function** $$\mathcal{Z}(\mu, V, T)$$ is defined as follows:
 
 $$\begin{aligned}
     \boxed{
@@ -52,9 +52,9 @@ $$\begin{aligned}
 
 In contrast to the canonical ensemble,
 whose [thermodynamic potential](/know/concept/thermodynamic-potential/)
-was the Helmholtz free energy $F$,
+was the Helmholtz free energy $$F$$,
 the grand canonical ensemble instead
-minimizes the **grand potential** $\Omega$:
+minimizes the **grand potential** $$\Omega$$:
 
 $$\begin{aligned}
     \boxed{
@@ -64,8 +64,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-So $\mathcal{Z} = \exp(- \beta \Omega)$.
-This is proven in the same way as for $F$ in the canonical ensemble.
+So $$\mathcal{Z} = \exp(- \beta \Omega)$$.
+This is proven in the same way as for $$F$$ in the canonical ensemble.
 
 
 
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