A PROBABILISTIC APPROACH FOR THE MEAN-FIELD LIMIT TO THE CUCKER-SMALE MODEL WITH A SINGULAR COMMUNICATION

摘要

We present a probabilistic approach for derivation of the kinetic Cucker-Smale (C-S) equation from the particle C-S model with singular communication. For the system we are considering, it is impossible to validate effective description for certain special initial data, thus such a probabilistic approach is the best one can hope for. More precisely, we consider a system in which kinetic trajectories are deviated from a microscopic model and use a suitable probability measure to quantify the randomness in the initial data. We show that the set of "bad initial data" does in fact have small measure and that this small probability decays to zero algebraically, as N tends to infinity. For this, we introduce an appropriate cut-off in the communication weight. We also provide a relation between the order of the singularity and the order of the cut-off such that the machinery for deriving classical mean-field limits introduced in [3] can be applied to our setting.