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Commit 705e8066 authored by Timon Idema's avatar Timon Idema
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Fixed typo in Maxwell dist plot legend.

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content/thermodynamicsbasics.md
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......@@ -414,7 +414,7 @@ v = np.linspace(0, 6, 600)
# Actual plots.
line1 = ax.plot(v, f(v, 1), label='low $T$')
line2 = ax.plot(v, f(v, 3), label='hight $T$')
line2 = ax.plot(v, f(v, 3), label='high $T$')
# Lines to plot from v-axis.
ax.plot([v0(1), v0(1)], [0, f(v0(1), 1)], color='C0', linestyle = '--')
......
notebooks/Maxwelldistribution.ipynb
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View file @ 705e8066
%% Cell type:markdown id: tags:
# Plot of the Maxwell distribution
For section 13.5.3
%% Cell type:markdown id: tags:
Maxwell distribution:
$$
f_\mathrm{Maxwell}(v) = \left( \frac{m}{2\pi k_\mathrm{B} T} \right)^{3/2} 4 \pi v^2 \exp\left(-\frac{m v^2}{2 k_\mathrm{B} T} \right)
$$
Average 'thermal speed' of each molecule
$$
v_\mathrm{th} = \sqrt{\frac{3 k_\mathrm{B} T}{m}}.
$$ (thermalspeed)
%% Cell type:code id: tags:
``` python
# Python libraries.
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
```
%% Cell type:code id: tags:
``` python
def f(v, T):
# Returns f(v), in units where kB = 1 and m = 1.
return np.power(1/(2*np.pi*T),1.5) * 4 * np.pi * v * v * np.exp(- 0.5 * v * v / T)
def v0(T):
# Returns v0(T), the maximum of the Maxwell distribution, in units where kB = 1 and m = 1.
return np.sqrt(2*T)
def vth(T):
# Returns v_{th}T, in units where kB = 1 and m = 1.
return np.sqrt(3*T)
```
%% Cell type:code id: tags:
``` python
fig, ax = plt.subplots(figsize=(6,4))
v = np.linspace(0, 6, 600)
line1 = ax.plot(v, f(v, 1), label='low $T$')
line2 = ax.plot(v, f(v, 3), label='hight $T$')
line2 = ax.plot(v, f(v, 3), label='high $T$')
# Lines to plot from v-axis.
ax.plot([v0(1), v0(1)], [0, f(v0(1), 1)], color='C0', linestyle = '--')
ax.plot([vth(1), vth(1)], [0, f(vth(1), 1)], color='C0', linestyle = ':')
ax.plot([v0(3), v0(3)], [0, f(v0(3), 3)], color='C1', linestyle = '--')
ax.plot([vth(3), vth(3)], [0, f(vth(3), 3)], color='C1', linestyle = ':')
# Below doesn't work as matplotlib interprets fractional values of ymax as fraction of total height.
#ax.axvline(x=v0(1), ymax=f(v0(1),1), color = 'k', linestyle = ':')
#ax.axvline(x=vth(1), ymax=f(vth(1), 1), color = 'k', linestyle = '--')
#ax.axvline(x=v0(3), ymax=f(v0(3), 3), color = 'k', linestyle = ':')
#ax.axvline(x=vth(3), ymax=f(vth(3), 3), color = 'k', linestyle = '--')
ax.set_xlim(0,6)
#ax.set_xticks([0, np.pi, 2*np.pi, 3*np.pi, 4*np.pi], labels=['$0$', '$\\pi$', '$2\\pi$', '$3\\pi$', '$4\\pi$'])
#ax.set_yticks([-1, -0.5, 0, 0.5, 1], labels=['$-1$', '$-\\frac{1}{2}$', '$0$', '$\\frac{1}{2}$', '$1$'])
ax.set_ylim(0,0.6)
ax.set_xlabel('$v$')
ax.set_ylabel('$f(v)$')
ax.text(v0(1), -0.01, r'$v_0$', color='C0', horizontalalignment='center', verticalalignment='top')
ax.text(vth(1), -0.01, r'$v_\mathrm{th}$', color='C0', horizontalalignment='center', verticalalignment='top')
ax.text(v0(3), -0.01, r'$v_0$', color='C1', horizontalalignment='center', verticalalignment='top')
ax.text(vth(3)+.18, -0.01, r'$v_\mathrm{th}$', color='C1', horizontalalignment='center', verticalalignment='top')
ax.legend()
#plt.savefig("dampedoscillations.pdf", format="pdf", bbox_inches="tight")
#plt.savefig("dampedoscillations.svg", format="svg", bbox_inches="tight")
```
%% Output
<matplotlib.legend.Legend at 0x12f52b680>
%% Cell type:code id: tags:
``` python
```
......
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