Diffusion maps for collective dynamics*

The emergence of coarse-grain dynamics is a prevalent feature in systems characterized by a separation of spatio-temporal scales. A classical example in Physics is thermodynamics, in which the evolution of the statistical moments of the particle ensembles offer a simpler description than the equations of motion of the particles themselves. In general, identifying coarse-grain variables allows us to simplify the description of a complex system and gain insights from analytical models. Finding the correct coarse-grain description is however challenging, particularly for living systems, since most of the intuition we have developed for classical physical systems does not apply. Here we show that a coarse-grain description of a high-dimensional dynamical system can be found by defining a measure of similarity between the different possible states that the system can achieve. In that sense, data mining techniques become suitable candidates for a systematic approach to find coarse grain descriptors. Here we show that the diffusion map technique is successful at finding the order parameters, identifying phase transitions and the mean-field descriptions of two model systems, the Ising model of ferromagnetism and the Vicsek model of flocking. We apply the same idea to biological data and identify collective states.