### 0.1 Drude-Lorentz Gas


**Definition 0.1.1 **  The **distribution function** denoted $f(\vec{r},\vec{p})$ for classical systems is the density of particle in position-momentum phase space. For $N$ particles at positions $\vec{r}_n$ and momenta $\vec{p}_n$, is given by the following sum of 3D Dirac deltas $\delta^3$.
\[f(\vec{r},\vec{p}) = \sum_{n=1}^N\delta^3(\vec{r}-\vec{r}_n)\delta^3(\vec{p}-\vec{p}_n)\]

**Definition 0.1.2 **  A **generic physical observable** $A$ **of distribution function** $f(\vec{r},\vec{p})$ is given the following integral where $A(\vec{r},\vec{p})$ is the contribution to that observable by a single particle in the distribution function.
\[A = \int d^3\vec{r}d^3\vec{p} f(\vec{r},\vec{p})A(\vec{r},\vec(p))\]

**Result 0.1.3 **  The **particle number** $N$ **of a distribution function** $f(\vec{r},\vec{p})$ is given by $N = \int d^3\vec{r}d^3\vec{p} f(\vec{r},\vec{p})$.

**Result 0.1.4 **  The **particle density** $n(\vec{r})$ **of a distribution function** $f(\vec{r},\vec{p})$ is given by $n(\vec{r}) = \int d^3\vec{p} f(\vec{r},\vec{p})$.

**Result 0.1.5 **  The **current density** $\vec{j}(\vec{r})$ **of a distribution function** $f(\vec{r},\vec{p})$ is given by $\vec{j}(\vec{r}) = \int d^3\vec{p} f(\vec{r},\vec{p}) \left(\frac{-e\vec{p}}{m}\right)$.

**Result 0.1.6 **  The **total energy** $U$ **of a distribution function** $f(\vec{r},\vec{p})$ is given by $U = \int d^3\vec{r}d^3\vec{p} f(\vec{r},\vec{p})$.

**Result 0.1.7 **  The **total energy** $\vec{p}$ **of a distribution function** $f(\vec{r},\vec{p})$ is given by $\vec{p} = \int d^3\vec{r}d^3\vec{p} f(\vec{r},\vec{p}) \vec{p}$.

**Definition 0.1.8 **  A **Drude-Lorentz gas** is a classical model of electrons in a solid that assume electrons relax to a Maxwell-Boltzmann equilibrium via simple random scattering with mean lifetime $\tau$. See the [supplementary note](https://kaedon.nethttps://kaedon.net/l/t4zf) for a guided derivation. The model describes the behavior of a distribution $f(\vec{r},\vec{p})$ of electrons with the following differential equation such that the equilibrium condition is the Maxwell-Boltzmann distribution $f_0(\vec{r},\vec{p})$.
\[\frac{d}{dt}f(\vec{r},\vec{p}) = \frac{f_0(\vec{r},\vec{p})-f(\vec{r},\vec{p})}{\tau}-\frac{\partial \vec{r}}{\partial t}\cdot\nabla_{\vec{r}}f(\vec{r},\vec{p}) - \frac{\partial \vec{p}}{\partial t}\cdot\nabla_{\vec{p}}f(\vec{r},\vec{p})\]
\[f_0(\vec{r},\vec{p}) = \frac{n e^{-\beta p^2/(2m)}}{(2\pi m_e k_B T)^{3/2}}\]

**Result 0.1.9 **  For weak fields $\vec{E}$ and $\vec\nabla T$, the distribution of the Drude-Lortenz Gas can be approximated by the following.
\[f(\vec{r},\vec{p}) \approx f_0 + \frac{\partial f}{\partial E}\vec{E} + \frac{\partial f}{\partial \vec\nabla T} \vec{\nabla}T\]

**Definition 0.1.10 **  The **linear response operators** denoted $L_{\alpha\beta}$ for $\alpha,\beta\in\{1,2\}$ describe the linear relationship for small electric fields $\vec{E}$ and temperature gradients $\vec\nabla T$ with the current $\vec{j}$ and heat current $\vec{j}_q$.
\[\begin{pmatrix} \vec{j}\\ \vec{j}_q \end{pmatrix} = \begin{pmatrix} L_{11} & L_{12} \\ L_{21} & L_{22} \end{pmatrix}\begin{pmatrix}    \vec{E} \\ -\vec{\nabla}T\end{pmatrix}\]

**Result 0.1.11 **  The **Drude-Lorentz electric conductivity** is the electric conductivity $\sigma$ predicted by the Drude-Lorentz model for small electric fields $\vec{E}$ and $\vec\nabla T =0$,
\[\vec{j} = \sigma \vec{E} = L_{11} \vec{E} = \frac{ne\tau}{m_e}\vec{E}\]

**Result 0.1.12 **  The **Drude-Lorentz heat conductivity** is the heat conductivity $\kappa$ predicted bt the Drude-Lorentz model for small temperature gradients $\vec\nabla T$ and $\vec{E} = 0$,
\[\vec{j}_q = -\kappa \vec\nabla T = -\left(L_{22} - \frac{L_{12}L_{21}}{L_{22}}\right)\vec\nabla T = -\frac{5}{2}\frac{k_B^2 nT\tau}{m_e}\vec\nabla T\]

**Result 0.1.13 **  The **Drude-Lorentz seebeck coefficient** is the seebeck coefficient $S$ predicted by the Drude-Lorentz model for small temperature gradients $\vec\nabla T$ and $\vec{j} = 0$,
\[\vec{E} = S\vec\nabla T = \frac{L_{12}}{L_{11}}\vec\nabla T = -\frac{k_B}{e} \vec\nabla T\]

**Result 0.1.14 **  The **Drude-Lorentz peltiercoefficient** is the peltier coefficient $\Pi$ predicted by the Drude-Lorentz model for small currents $\vec{j}\neq 0$ and $\vec\nabla T = 0$,
\[\vec{j}_q = \Pi \vec{j} = ST\vec{j} = \frac{k_BT}{e}\vec{j}\]

**Result 0.1.15 **  The **Drude-Lorentz molar heat capacity** is the molar heat capacity $c_{\text{el,mol}}$ predicted by the Drude-Lorentz model,
\[c_{\text{el,mol}} = \frac{3}{2} R Z\]

**Law 0.1.16 **  The **Drude-Lorentz Wiedemann–Franz law** describes the [Lorentz Number](https://kaedon.net/l/217r) predicted by the Drude-Lorentz model.
\[L = \frac{\kappa}{\sigma L} = \frac{5}{2}\left( \frac{k_B}{e} \right)^2\]

**Note 0.1.17 **  While the Drude-Lorentz gas successfully predicts the existence of these effects, they value of the coefficients are off by orders of magnitude.