Autonomous Mobile Robots and Multi-Robot Systems

12Brownian Motion and Swarm Dynamics

Eugene Khmelnitsky

12.1 Langevin and Fokker‐Plank Formalism

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:

(12.1)

where ξ(t) is a Gaussian noise, i.e. the values of ξ(t) at any pair of times are identically distributed, ξ(t)~N(0,σ²), and uncorrelated F(x,t) represents additional forces (Sjögren 2015, Schimansky‐Geier et al. 2005).

Since the dynamics are random, we are interested in the probability density p(x, v, t) to find the Brownian particle in the interval (x, x + dx) and (v, v + dv) at time t. The derivation of the dynamic equation for p(x, v, t) – the Fokker‐Planck equation – is carried out in two steps. The first one derives the equation of motion for the probability density ρ(x, v, t) for one realization of the random force ξ(t). The second step averages the value of ρ(x, v, t) over many realizations:

For simplicity, we derive the Fokker‐Planck equation, assuming no external fields, i.e. F(x, t) = 0.

The Brownian particle is located ...