We consider a minimalist model of overtaking dynamics in one dimension. On each site of a one-dimensional infinite lattice sits an agent carrying a random number specifying the agent's preferred velocity, which is drawn initially for each agent independently from a common distribution. The time evolution is Markovian, where a pair of agents at adjacent sites exchange their positions with a specified rate, while retaining their respective preferred velocities, only if the preferred velocity of the agent on the “left” site is higher. We discuss two different cases: one in which a pair of agents at sites $i$ and $i+1$ exchange their positions with rate 1, independent of their velocity difference, and another in which a pair exchange their positions with a rate equal to the modulus of the velocity difference. In both cases, we find that the net number of overtake events by a tagged agent in a given duration $t$, denoted by $m\left(t\right)$, increases linearly with time $t$, for large $t$. In the first case, for a randomly picked agent, $m/t$, in the limit $t\to \infty$, is distributed uniformly on $[-1,1]$, independent of the distributions of preferred velocities. In the second case, the distribution is given by the distribution of the preferred velocities itself, with a Galilean shift by the mean velocity. We also find the large time approach to the limiting forms and compare the results with numerical simulations.

• Received 17 April 2018
• Revised 5 October 2018

DOI:https://doi.org/10.1103/PhysRevE.98.052122