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In this article we study a system of $N$ particles, each of them beingdefined by the couple of a position (in $\mathbb{R}^d$) and a so-calledorientation which is an element of a compact Riemannian manifold. Thisorientation can be seen as a generalisation of the velocity in Vicsek-typemodels such as [Degond et al. 2017; Degond, Motsch 2008]. We will assume thatthe orientation of each particle follows a jump process whereas its positionevolves deterministically between two jumps. The law of the jump depends on theposition of the particle and the orientations of its neighbours. In the limit$N\to\infty$, we first prove a propagation of chaos result which can be seen asan adaptation of the classical result on McKean-Vlasov systems [Sznitman 1991]to Piecewise Deterministic Markov Processes (PDMP). As in [Jourdain,M\'el\'eard 1998], we then prove that under a proper rescaling with respect to$N$ of the interaction radius between the agents (moderate interaction), thelaw of the limiting mean-field system satisfies a BGK equation with localisedinteractions which has been studied as a model of collective behaviour in[Degond et al. 2018]. Finally, in the spatially homogeneous case, we give analternative approach based on martingale arguments.

Propagation of chaos and moderate interaction for a piecewise deterministic system of geometrically enriched particles