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 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 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.