diff --git a/src/3_drude_model.md b/src/3_drude_model.md index b797c2685be92a9729a585c33807d160474e9bde..02cd6a08163e126f6d800b9a884e0afbc7225cd2 100644 --- a/src/3_drude_model.md +++ b/src/3_drude_model.md @@ -25,6 +25,45 @@ Ohm's law states that $V=IR=I\rho\frac{l}{A}$. In this lecture we will investiga - At each scattering event an electron returns to momentum ${\bf p}=0$. - In-between scattering events electrons respond to the Lorentz force ${\bf F}_{\rm L}=-e\left({\bf E}+{\bf v}\times{\bf B}\right)$. +```python +import numpy as np +import matplotlib.pyplot as plt + +walker_number = 20 # number of particles +tau = 1 # relaxation time +gamma = .3 # dissipation strength +a = 1 # acceleration +dt = .1 # infinitesimal +T = 20 # simulation time + +v = np.zeros((2, int(T // dt), walker_number), dtype=float) + +scattering_events = np.random.binomial(1, dt/tau, size=v.shape[1:]) +angles = np.random.uniform(high=2*np.pi, size=scattering_events.shape) * scattering_events +rotations = np.array( + [[np.cos(angles), np.sin(angles)], + [-np.sin(angles), np.cos(angles)]] +) + +for step in range(1, v.shape[1]): + v[:, step] = v[:, step-1] + v[0, step] += a * dt + v[:, step] = np.einsum( + 'ijk,jk->ik', + rotations[:, :, step-1, :], + v[:, step, :] + ) * (1 - gamma * scattering_events[step-1]) + +r = np.cumsum(v * dt, axis=1) + +scattering_positions = np.copy(r) +scattering_positions[:, ~scattering_events.astype(bool)] = np.nan + +plt.plot(*r[:, :100], alpha=.5, c='#1f77b4'); +plt.scatter(*scattering_positions[:, :100], s=10); +plt.axis('off'); +``` + We start by considering only an electric field (_i.e._ ${\bf B}=0$). What velocity do electrons acquire in-between collisions? $$ @@ -124,4 +163,4 @@ $$\mathbf{E} = \begin{pmatrix} \rho_{xx} & \rho_{xy} \\ \rho_{yx} & \rho_{yy} \e 2. Invert the resistivity matrix to obtain the conductivity matrix $$\begin{pmatrix} \sigma_{xx} & \sigma_{xy} \\ \sigma_{yx} & \sigma_{yy} \end{pmatrix} $$, allowing you to express $\mathbf{J}$ as a function of $\mathbf{E}$. 3. Sketch $\sigma_{xx}$ and $\sigma_{xy}$ as a function of the magnetic field $\bf B$. 4. Give the definition of the Hall coefficient. What does the sign of the Hall coefficient indicate? - \ No newline at end of file +