# Homework 5¶

**due 10/17/2018 at 11:59pm**

## Problem 1¶

Using the binary collision function of HW03, plot the distribution function of 25 protons and 25 electrons locked inside a box of size 2mmx2mm, randomly distributed, for each of the following initial parameters:

- $v_i=v_e=1000\,m/s$, all with random directions
- $v_i=v_e/10=100\,m/s$, all with random directions

Note that we will suppose that there is no displacement along the Z-direction and that all particles reflect back inside the box when they hit the walls of the box.

## Problem 2¶

Compute numerically the total number of particles for an infinitely large box where the particles inside have the following distribution function:
$$f(\vec x,\vec v,t)=A_0\exp\Big(-\frac{x^2+y^2+z^2}{L_0^2}\Big)\exp\Big(-\frac{v_x^2+v_y^2+v_z^2}{v_0^2}\Big)$$
where $A_0=200\,particles/m^6s^3$, $L_0=2 m$ and $v_0=0.1\,m/s$. Compare the numerical answer with the analytical answer using the fact that $$\int_{-\infty}^{+\infty}\exp(-x^2)dx=\sqrt{\pi}.$$