Brownian Motion and Swarm Dynamics

This chapter considers the model of the swarm dynamics by the use of Langevin and Fokker-Plank equations. These equations allow the description the swarm motion using the methods of statistical physics. The interactions between the environment and the Brownian particle results in three forces that act on the Brownian particle: a damping force, a force associated with the exogenous fields, and a fluctuating force. The forces affect the dynamics of the location and speed of the Brownian particle, as given in the one-dimensional Langevin equation. The dynamics of speed follow the Brownian motion, growing linearly with time. The chapter presents examples to show that the presence of the damping force restricts the motion in such a way that the dependence of the location variance on time drops down from a power of three to linearly proportional, and the speed variance loses its dependence on time at all.