Improve knowledge baseHEADmaster

1 files changed, 19 insertions, 19 deletions

diff --git a/source/know/concept/fermi-dirac-distribution/index.md b/source/know/concept/fermi-dirac-distribution/index.md
index 2a38eb3..7554e5a 100644
--- a/source/know/concept/fermi-dirac-distribution/index.md
+++ b/source/know/concept/fermi-dirac-distribution/index.md
@@ -13,18 +13,18 @@ layout: "concept"
 which obey the [Pauli exclusion principle](/know/concept/pauli-exclusion-principle/),
 distribute themselves across the available states in a system at equilibrium.
 
-Consider one single-particle state $$s$$,
+Consider one single-particle state $$\ket{i}$$,
 which can contain $$0$$ or $$1$$ fermions.
-Because the occupation number $$N$$ is variable,
+Because the occupation number $$n_i$$ is variable,
 we turn to the [grand canonical ensemble](/know/concept/grand-canonical-ensemble/),
 whose grand partition function $$\mathcal{Z}$$ is as follows,
-where $$\varepsilon$$ is the energy of $$s$$
+where $$\varepsilon_i$$ is the energy of $$\ket{i}$$
 and $$\mu$$ is the chemical potential:
 
 $$\begin{aligned}
     \mathcal{Z}
-    = \sum_{N = 0}^1 \Big( e^{-\beta (\varepsilon - \mu)} \Big)^N
-    = 1 + e^{-\beta (\varepsilon - \mu)}
+    = \sum_{m = 0}^1 \Big( e^{-\beta (\varepsilon_i - \mu)} \Big)^m
+    = 1 + e^{-\beta (\varepsilon_i - \mu)}
 \end{aligned}$$
 
 The corresponding [thermodynamic potential](/know/concept/thermodynamic-potential/)
@@ -33,43 +33,43 @@ is the Landau potential $$\Omega$$, given by:
 $$\begin{aligned}
     \Omega
     = - k T \ln{\mathcal{Z}}
-    = - k T \ln\!\Big( 1 + e^{-\beta (\varepsilon - \mu)} \Big)
+    = - k T \ln\!\Big( 1 + e^{-\beta (\varepsilon_i - \mu)} \Big)
 \end{aligned}$$
 
-The average number of particles $$\expval{N}$$
-in $$s$$ is then found by taking a derivative of $$\Omega$$:
+The average number of particles $$\expval{n_i}$$
+in $$\ket{i}$$ is then found by taking a derivative of $$\Omega$$:
 
 $$\begin{aligned}
-    \expval{N}
+    \expval{n_i}
     = - \pdv{\Omega}{\mu}
     = k T \pdv{\ln{\mathcal{Z}}}{\mu}
-    = \frac{e^{-\beta (\varepsilon - \mu)}}{1 + e^{-\beta (\varepsilon - \mu)}}
+    = \frac{e^{-\beta (\varepsilon_i - \mu)}}{1 + e^{-\beta (\varepsilon_i - \mu)}}
 \end{aligned}$$
 
-By multiplying both the numerator and the denominator by $$e^{\beta (\varepsilon \!-\! \mu)}$$,
+By multiplying both the numerator and the denominator by $$e^{\beta (\varepsilon_i - \mu)}$$,
 we arrive at the standard form of
 the **Fermi-Dirac distribution** or **Fermi function** $$f_F$$:
 
 $$\begin{aligned}
     \boxed{
-        \expval{N}
-        = f_F(\varepsilon)
-        = \frac{1}{e^{\beta (\varepsilon - \mu)} + 1}
+        \expval{n_i}
+        = f_F(\varepsilon_i)
+        = \frac{1}{e^{\beta (\varepsilon_i - \mu)} + 1}
     }
 \end{aligned}$$
 
-This gives the expected occupation number $$\expval{N}$$
-of state $$s$$ with energy $$\varepsilon$$,
+This gives the expected occupation number $$\expval{n_i}$$
+of state $$\ket{i}$$ with energy $$\varepsilon_i$$,
 given a temperature $$T$$ and chemical potential $$\mu$$.
 
 {% comment %}
-The corresponding variance $$\sigma^2 \equiv \expval{N^2} - \expval{N}^2$$ is found to be:
+The corresponding variance $$\sigma^2 \equiv \expval{n_i^2} - \expval{n_i}^2$$ is found to be:
 
 $$\begin{aligned}
     \boxed{
         \sigma^2
-        = k T \pdv{\expval{N}}{\mu}
-        = \expval{N} \big(1 - \expval{N}\big)
+        = k T \pdv{\expval{n_i}}{\mu}
+        = \expval{n_i} \big(1 - \expval{n_i}\big)
     }
 \end{aligned}$$
 {% endcomment %}