Random Averaging

Eli Ben-Naim
Los Alamos National Laboratory

Averaging processes underlie collisions in granular media as well interactions of social and opinion dynamics. In the averaging process there are are infinitely many interacting "particles", each characterized by a single variable. In the averaging process, two particles are chosen at random and both are set to their average. We study averaging processes using kinetic theory and find a number of interesting phenomena:

1) Multiscaling. The moments of the distribution exhibit multiscaling so that knowledge of the average behavior is not sufficient to characterize the probability distribution function.

2) Extremal selection. The characteristic scale may obey an extremal selection principle, as in nonlinear traveling waves.

3) Patterns and bifurcations. When the averaging process excludes particles that exceeds a threshold, there system organizes into clusters. These clusters are patterned and the number of clusters undergoes a series of bifurcations as a function of the initial conditions.

4) Synchronization. When the averaged quantity represents a phase and in the presence of noise, there is phase transition from an ordered state where the particles are aligned into a disordered state where the particles are not correlated. The behavior is nicely related to a special partition of the integers.

Audio (MP3 File, Podcast Ready) Presentation (PDF File)

Back to Workshop II: Random Curves, Surfaces, and Transport