Then, how can we theoretically understand the Lévy flight dynamics as statistical physics? Unfortunately, this question has not been well discussed from its underlying particle dynamics. Contrary to the Brownian motion, most of the previous studies on the Lévy flight dynamics is based on experimental fitting or phenomenological discussion; its systematic derivation from the underlying particle dynamics has been still missing. In this talk, we will present a statistical-physics theory to derive the Lévy flight model for an active matter system from its particle dynamics.
We consider a dynamical system composed of swimming organisms. Here, the swimmers are assumed to exhibit unidirectional motion, inducing hydrodynamic flow on the the passive tracer dynamics. Considering the long-range nature of the hydrodynamics interaction, we have developed a non-Markovian kinetic theory for the tracer dynamics in this active suspension. We theoretically show that the tracer exhibits power-law displacement statistics and a crossover in its power-law exponents between short and long timescales. Furthermore, we have shown that the coarse-graining description of this model at the longer timescale corresponds to the Lévy flight model effectively.