The momentum distribution of a one-dimensional ideal gas of N atoms

We calculate the average momentum distribution and the average deviation from this distribution for a one-dimensional ideal gas of N atoms. The calculation is performed using analytical and numerical methods, and the results obtained using the two methods are compared. We use these results to show that in the limit of large N, almost all the possible microstates of an ideal gas have momentum distributions that are very close to the Maxwell distribution. We discuss the significance of this fact for understanding why an ideal gas approaches thermal equilibrium.