Explicit decay rate for the Gini index in the repeated averaging model

We investigate the repeated averaging model for money exchanges: two agents picked uniformly at random share half of their wealth to each other. It is intuitively convincing that a Dirac distribution (centered initial average wealth) will be long time equilibrium this dynamics. In other words, Gini index should converge zero. To better understand dynamics, we its limit as number goes infinity by proving so-called propagation chaos, which links stochastic agent-based dynamics (limiting) nonlinear partial differential equation (PDE). This deterministic description has flavor classical Boltzmann arising from statistical mechanics dilute gases. prove convergence towards showing associated converges zero with an explicit rate.