  We consider a stochastic $N$-particle system on a torus in which each
particle moving freely can instantaneously thermalize according to the particle
configuration at that instant. Following [2], we show that the propagation of
chaos does hold and that the one-particle distribution converges to the
solution of the BGK equation. The improvement with respect to [2] consists in
the fact that here, as suggested by physical considerations, the thermalizing
transition is driven only by the restriction of the particle configuration in a
small neighborhood of the jumping particle. In other words, the Maxwellian
distribution of the outgoing particle is computed via the empirical
hydrodynamical fields associated to the fraction of particles sufficiently
close to the test particle and not, as in [2], via the whole particle
configuration.
  [2]: arXiv:2002.10535 (Journal reference: Arch. Ration. Mech. Anal. Vol. 240
(2021), pp. 785-808)
