Modeling flocks and prices: jumping particles with an attractive interaction (Joint work with Miklos Racz and Balint Toth)

Stochastic Analysis Seminar Series


I will introduce a model of a finite number of competing particles on R.

Real-life phenomena that could be modeled this way possibly include the evolution of stocks in a market, or herding behavior of animals. Given a particle configuration, the center of mass of the particles is computed by simply averaging the particle locations. The evolution is a continuous time Markov jump process: given a configuration and thus the center of mass, each particle jumps with a rate that depends on the particle's relative position compared to the center of mass. Those left behind have a higher jump rate than those in front of the center of mass. When a jump of a particle occurs, the jump length is chosen independently of everything from a positive distribution. Hence we see that the dynamics tries to keep the particles together.

The main point of interest is the behavior of the model as the number of particles goes to infinity. We first heuristically wrote up a differential equation on the evolution of particle density. I will explain the heuristics, and show travelling wave solutions in a few cases. I will also present a surprising connection to extreme value statistics. Then I will briefly sketch a hydrodynamic argument which proves that the evolution of the system indeed converges to that governed by the differential equation.




Monday, March 10, 2014 - 15:45
to 16:45