Random walk of a “drunk company”

Abstract

We study the collective behavior of a system of Brownian agents each of which moves orienting itself to the group as a whole. This system is the simplest model of the motion of a “united drunk company.” For such a system, we use the functional integration technique to calculate the probability of transition from one point to another and to determine the time dependence of the probability density to find a member of the “drunk company” near a given point. It turns out that the system exhibits an interesting collective behavior at large times and this behavior cannot be described by the simplest mean-field-type approximation. We also obtain an exact solution in the case where one of the agents is “sober” and moves along a given trajectory. The obtained results are used to discuss whether such systems can be described by different theoretical approaches.