### 0.1 Drude-Lorentz Gas


**Definition 0.1.1 **  The **temperature gradient** denoted $\vec{\nabla} T(\vec{r})$ is the the gradient of temperature $T(\vec{r})$ at position $\vec{r}$ in a material.

**Definition 0.1.2 **  The **heat current** denoted $\vec{j}_q$ is the rate of energy transfer through a material due to temperature gradient.

**Definition 0.1.3 **  The **thermal conductivity** denoted $k$ of a material is the coefficient that relates the temperature gradient $\nabla T$ to the heat current $\vec{j}_q$.
\[\vec{j}_q = -k\sigma \vec{E}\]

**Definition 0.1.4 **  

**Definition 0.1.5 **  

**Result 0.1.6 **  

**Result 0.1.7 **  

**Result 0.1.8 **  

**Result 0.1.9 **  

**Result 0.1.10 **  

**Definition 0.1.11 **  \[\text{DYNAMIC EQUATION}\]
\[\text{Equilibrium Condition}\]

**Definition 0.1.12 **  linear response coefficients

**Result 0.1.13 **  electric conductivity, heat conductivity, seabeck

**Result 0.1.14 **  Drude-Lorentz Heat capacity

**Definition 0.1.15 **  The **Lorentz number** denoted $L$ is the proportionality constant that relates the thermal conductivity $\kappa$ to the electric conductivity $\sigma$ at temperature $T$.
\[L = \frac{\kappa}{\sigma T}\]

**Law 0.1.16 **  The **Wiedemann–Franz law** states that the [Lorentz number](https://kaedon.net/l/217r) $L$ is a constant that can be written in terms of the Boltzmann constant $k_B$ and the elementary charge $e$.
\[L = \frac{\kappa}{\sigma L} = \frac{\pi^2}{3}\left( \frac{k_B}{e} \right)^2\]