td-wslda-examples.md

+# Example 1: Time-dependent external potential
+Target: find a time evolution of the unitary Fermi gas confined in a harmonic trap:
+```math
+V_{\textrm{ext}}(x,y,z,t) = \frac{m\omega_x^2 x^2}{2} + A(t)\frac{m\omega_y^2 y^2}{2}
+```
+where $`A(t)`$ is a function that smoothly rises from zero to one within a given time interval.
+
+## Step 1: Find the static solution 
+The initial external potential is $`V_{\textrm{ext}}(x,y,z) = \frac{m\omega_x^2 x^2}{2}`$, and we can use 1d code.
+
+Code: `st-wslda-1d`
+
+Settings:
+* [trap-1d_predefines.h](uploads/fa2a85fae8ae389f5856b11ab0d63ca5/trap-1d_predefines.h)
+* [trap-1d_problem-definition.h](uploads/4dbed95bba4ab63fd9d109993b1da1dc/trap-1d_problem-definition.h)
+* [trap-1d_logger.h](uploads/674dcdf9f8827ade7b0dcae0aafb124a/trap-1d_logger.h)
+* [trap-1d_input.txt](uploads/a3660b6b3e7397ca478080818e56ab90/trap-1d_input.txt)
+
+Output:
+* [trap-1d.stdout](uploads/30ac8d3313e93f3554505a081f186cf1/trap-1d.stdout)
+
+## Step 2: Execute time evolution
+
+Code: `td-wslda-2d`
+
+Settings:
+* [trap-2d-alpha0.0_predefines.h](uploads/e3cf4756f7c5ef870d36dab31def1914/trap-2d-alpha0.0_predefines.h)
+* [trap-2d-alpha0.0_problem-definition.h](uploads/59850de149fd98c21b8575d3a1514f30/trap-2d-alpha0.0_problem-definition.h)
+* [trap-2d-alpha0.0_logger.h](uploads/0a334f058908f7f0ea0b1b1e15ee4d10/trap-2d-alpha0.0_logger.h)
+* [trap-2d-alpha0.0_input.txt](uploads/9edac616780f9196c9ccf3253fedc6a9/trap-2d-alpha0.0_input.txt)
+
+Output:
+* [trap-2d-alpha0.0.stdout](uploads/3221116b30a9e6cb500544f943a4e5e1/trap-2d-alpha0.0.stdout)
+* [trap-2d-alpha0.0.wlog](uploads/cc008e14db1172dd5e7c9df0b7c63472/trap-2d-alpha0.0.wlog)
+
+![td-snapshots](uploads/baa3c05d97b6c92f57aa97f9f6d3dec4/td-snapshots.png)
+
+Energy evolution:
+```python
+import numpy as np
+import matplotlib.pyplot as plt
+
+data = np.loadtxt("/home/gabrielw/sshfs/lumi/scratch/quantum_friction/td-ho-gw/trap-2d-alpha0.0.wlog", usecols=(1,4,5))
+
+fig, ax = plt.subplots()
+ax.plot(data[:,0], data[:,2], color='red', label=r'energy', lw=3.0)
+ax.set(xlabel=r'$t\varepsilon_F$', ylabel=r'$E/E_{ffg}$')
+ax2 = ax.twinx()  # instantiate a second axes that shares the same x-axis
+ax2.plot(data[:,0], data[:,1], color='blue', label=r'particle number', lw=2.0, ls="--") # plot last frame [-1]
+ax2.set(ylabel=r'$N$')
+fig.legend(loc="upper left", bbox_to_anchor=(0.3,0.3), bbox_transform=ax.transAxes)
+fig.savefig("energy-conservation.png")
+```
+
+![energy-conservation](uploads/9dda79ee393867e04648651d71783ed5/energy-conservation.png)
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