%0 Journal Article
%T Solvable model of the collective motion of heterogeneous particles interacting on a sphere
%A Takuma Tanaka
%J Physics
%D 2014
%I arXiv
%R 10.1088/1367-2630/16/2/023016
%X I propose a model of mutually interacting particles on an M-dimensional unit sphere. I derive the dynamics of the particles by extending the dynamics of the Kuramoto-Sakaguchi model. The dynamics include a natural-frequency matrix, which determines the motion of a particle with no external force, and an external force vector. The position (state variable) of a particle at a given time is obtained by the projection transformation of the initial position of the particle. The same projection transformation gives the position of the particles with the same natural-frequency matrix. I show that the motion of the centre of mass of an infinite number of heterogeneous particles whose natural-frequency matrices are obtained from a class of multivariate Lorentz distribution is given by an M-dimensional ordinary differential equation in closed form. This result is an extension of the Ott-Antonsen theory.
%U http://arxiv.org/abs/1401.3109v1