Homework 1
1.) Mawell Boltzmann
Given the Maxwell -Boltzmann Distribution
- [math]N(v) = 4 \pi \left ( \frac{m}{2\pi kT}\right)^{3/2} v^2 e^{-\frac{mv^2}{2kT}}[/math]
a.) Show <v>
Show that
- [math]\lt v\gt = 4\pi \left ( \frac{m}{2 \pi kT}\right )^{3/2} \left( \frac{2kT}{m}\right)^2 \frac{\Gamma(2)}{2}[/math]
b.) Energy Fluctuation
Show that the energy fluctuation is
- [math]\frac{1}{4} m \lt \left ( v^2 - \lt v^2\gt \right)^2\gt = \frac{3}{2} (kT)^2[/math]
- Note
- [math]\lt \left ( v - \lt v\gt \right)^2\gt = \lt v^2 - 2v\lt v\gt + \lt v\gt ^2\gt = \lt v^2\gt - (\lt v\gt )^2[/math]
- = [math]\frac{3kT}{m} - \frac{8kT}{m}[/math] = velocity fluctuation
- [math]\frac{m^2}{4} \lt \left ( v^2 - \lt v^2\gt \right)^2\gt = \frac{m}{4}\left ( \lt v^4\gt - (\lt v^2\gt )^2 \right )[/math]
- =\frac{1}{4} \left ( 15(kT)^2 - (3kT)^2\right)
2.) MC calculation of Pi
Calculate \pi using the Monte Carlo method described in the Notes
3.) Histograms using ROOT
Homework 2
Homework 3
Homework 4
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