Statistics of overtake events by a tagged agent

Abstract

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(t), increases linearly with time t, for large t. In the first case, for a randomly picked agent, m/t, in the limit t→∞, 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.