Dynamics of hard rods in one dimension

Abstract

We examine a system consisting ofN classical, Newtonian, perfectly elastic hard rods constrained to move on a line. The mass and length of each rod are arbitrary. We develop an algorithm which gives, after any given possible sequence of collisions, the new velocities of theN rods and a necessary condition for any given pair of rods to be involved in the next collision, all in terms of the initial velocities of the rods. These results are then used to prove that for the case where there are exactly three rods on the line, the maximum possible number of collisions among them is the largest integern such that$m_2< (\mu _{12} \mu _{23} )^{1/2} /\cos \left[ {\pi /(n - 1)} \right]$, wherem ₂ is the mass of the central particle andμ ₁₂ andμ ₂₃ are the reduced masses of the left and right particle pairs. We further derive for this three-particle case a condition on the initial velocities which is necessary and sufficient fork collisions, 1<k≤n, to occur, as well as explicit expressions for the velocities after each collision in terms of the initial velocities.