Propagation of chaos: A review of models, methods and applications. Ⅱ. Applications

Figure 1.  An interaction graph. The vertical axis represents time. Each particle is represented by a vertical line parallel to the time axis. The index of a given particle is written on the horizontal axis. The construction is done backward in time starting from time $ t $ where only particle $ i $ is present. At each time $ t_\ell $, if $ i_\ell $ does not already belong to the graph, it is added on the right (with a vertical line which starts at $ t_\ell $). The couple $ r_\ell = (i_\ell,j_\ell) $ of interacting particles at time $ t_\ell $ is depicted by an horizontal line joining two big black dots on the vertical line representing the particles $ i_\ell $ and $ j_\ell $. for instance, on the depicted graph, $ r_2 = (i_2,i) $. Note that at time $ t_3 $, $ r_3 = (i_1,i_2) $ (or indifferently $ r_3 = (i_2,i_1) $) where $ i_1 $ and $ i_2 $ were already in the system. Index $ i_3 $ is skipped and at time $ t_4 $, the route is $ r_4 = (i_4,i_1) $. The recollision occurring at time $ t_3 $ is depicted in red

Figure 2.  A route of size $ q $ between $ i $ and $ j $. The chain of interactions which links $ i $ and $ j $ are depicted by horizontal lines as explained in Section 2.3.5

Figure 3.  At time $ t_1^+ $, there is only one particle, at the same position for the BBGKY (in white) and Boltzmann (in red) pseudo-trajectories. At time $ t_1^+ $, a particle (dotted) is added next to the white particle in a pre-collisional way. At time $ t_2^+ $, a particle (dashed) is added next to the dotted particle in a post-collisional way. Due to a recollision at time $ \tau\in(t_3,t_2) $, the Boltzmann and BBGKY pseudo-trajectories of the white/red particle are no longer close to each other at time $ t_3 $