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Modeling Flocks and Prices: Jumping Particles with an Attractive Interaction

Mathematical Physics & Probability

Speaker: Miklos Racz, UC Berkeley
Location: 1147 MSB
Start time: Wed, Nov 14 2012, 4:10PM

We introduce and investigate a model of a finite number of competing particles jumping forward on the real line. The evolution of the particles is a continuous-time Markov jump process: given a configuration, each particle jumps with a rate that depends on the particle's relative position compared to the center of mass of the system. The rates are higher for those left behind, and lower for those ahead of the center of mass, providing an attractive interaction keeping the particles together. Whenever a jump occurs, the jump length is chosen independently of everything else from a positive distribution. Real-life phenomena that could be modeled this way includes the evolution of prices in a market, or herding behavior of animals. The main point of interest is the behavior of the model as the number of particles goes to infinity. We prove that in this fluid limit the evolution of the system is described by a mean field equation that exhibits traveling wave solutions. We also present a surprising connection to extreme value statistics. This is joint work with Marton Balazs and Balint Toth.