Exercise 4.4: Distribution of speeds

Our goal is to calculate the equilibrium probability $P(v)\Delta v$ of finding a particle with speed between $v$ and $v+\Delta v$. Obtain an estimate of the maximum speed of the particles from the initial configuration. Choose windows with width $\Delta v$. Each particle will belong to the $k$-th window, with $k=v/\Delta v$. A reasonable choice of $% \Delta v$ is $0.1\sqrt{T}$, where $T$ is the temperature.

1. Record the histogram of velocities $P\left( v\right)$ and plot is versus $v$. What is the qualitative form of $P(v)$? What is the most probable value for $v$? What is the approximate ``width'' of $P(v)$? This probability distribution is known as the Maxwell-Boltzman distribution.

2. Determine the distribution for each component of the velocity. Make sure to distinguish between positive and negative values. What are the most probable values for the $x$ and $y$ components? What are the average values?