Kinematrix Theory and Universalities in Self-Prolpellers, Swimmers and Nanomotors

We present an elegant and efficient matrix-based theory for modeling systems within the emerging field of self-propellers, autonomous movers, and active swimmers as an alternative to traditional differential-equation-based Fokker-Planck and Langevin formalisms for the class of stochastic processes with negligible correlation time. Such self-propellers range from micron and submicron biological and artificial motors to macroscale animals, swimmers and pedestrians. The motions of self-propellers naturally decompose into elementary processes such as deterministic linear and rotational motion, stochastic orientational diffusion, flipping, and tumbling. While traditional differential-equation based formalisms become cumbersome as the number of elementary processes in a model increases, our theory treats all the elementary processes on the same footing. A matrix, called kinematrix, is produced from inspection of the elementary processes and from it many ensemble properties of a self-propeller such as autocorrelations of linear and angular velocities, mean-square displacement, and effective diffusion are derived by simple matrix algebra. We examine four different classes of self-propellers and show that the kinematrix formalism can explain the behavior of these systems in a way that reveals underlying universalities previously unrecognized within the more cumbersome traditional treatments.