Interacting Brownian Swarms: Some Analytical Results

Interacting Brownian Swarms: Some Analytical Results

We consider the dynamics of swarms of scalar Brownian agents subject to local imitation mechanisms implemented using mutual rank-based interactions. For appropriate values of the underlying control parameters, the swarm propagates tightly and the distances separating successive agents are iid exponential random variables. Implicitly, the implementation of rank-based mutual interactions, require...

Swarms with canonical active Brownian motion.

We present a swarm model of Brownian particles with harmonic interactions, where the individuals undergo canonical active Brownian motion, i.e., each Brownian particle can convert internal energy to mechanical energy of motion. We assume the existence of a single global internal energy of the system. Numerical simulations show amorphous swarming behavior as well as static configurations. Analyt...

Soft Control of Swarms - Analytical Approach

We analytically study the collective dynamics of mutually interacting heterogeneous agents evolving in a random environment. Our formal framework consist of a collection of N scalar drifted Brownian motions (BM) diffusing on R. The mutual interactions are introduced via a ranked-based, real-time mechanism always endowing the laggard, (i.e the agent with the leftmost position), with an extra pos...

Some new results on Brownian Directed Polymers in Random Environment

We prove some new results on Brownian directed polymers in random environment recently introduced by the authors. The directed polymer in this model is a d-dimensional Brownian motion (up to finite time t) viewed under a Gibbs measure which is built up with a Poisson random measure on R+×R (=time × space). Here, the Poisson random measure plays the role of the random environment which is indepe...

Interacting Brownian particles with strong repulsion