Random walk of a "drunk company"

A. G. Semenov<sup>abc

a</sup> Tamm Department of Theoretical Physics, Lebedev Physical Institute, RAS, Moscow, Russia
^b National Research University Higher School of Economics, Moscow, Russia
^c Russian Endowment for Education and Science, Moscow, Russia

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.

Keywords: Brownian agent, stochastic dynamics, functional integration method.

Received: 25.08.2015

DOI: 10.4213/tmf9033

 English version: Theoretical and Mathematical Physics, 2016, 187:2, 753–761

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