Difference between revisions of "HomeWork Simulations of Particle Interactions with Matter"

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:= <math>\frac{3kT}{m} - \frac{8kT}{m}</math> = velocity fluctuation
 
:= <math>\frac{3kT}{m} - \frac{8kT}{m}</math> = velocity fluctuation
 
: <math>\frac{m^2}{4} < \left ( v^2 - <v^2>\right)^2> = \frac{m}{4}\left ( <v^4> - (<v^2>)^2 \right )</math>  
 
: <math>\frac{m^2}{4} < \left ( v^2 - <v^2>\right)^2> = \frac{m}{4}\left ( <v^4> - (<v^2>)^2 \right )</math>  
:=\frac{1}{4} \left ( 15(kT)^2 - (3kT)^2\right)
+
:<math>=\frac{1}{4} \left ( 15(kT)^2 - (3kT)^2\right)</math>
  
 
==2.) MC calculation of Pi==
 
==2.) MC calculation of Pi==

Revision as of 21:12, 27 September 2007

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]
[math]=\frac{1}{4} \left ( 15(kT)^2 - (3kT)^2\right)[/math]

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

Back to Notes