  We investigate the unbiased model for money exchanges: agents give at random
time a dollar to one another (if they have one). Surprisingly, this dynamics
eventually leads to a geometric distribution of wealth (shown empirically by
Dragulescu and Yakovenko in [11] and rigorously in [2,12,15,18]). We prove a
uniform-in-time propagation of chaos result as the number of agents goes to
infinity, which links the stochastic dynamics to a deterministic infinite
system of ordinary differential equations. This deterministic description is
then analyzed by taking advantage of several entropy-entropy dissipation
inequalities and we provide a quantitative almost-exponential rate of
convergence toward the equilibrium (geometric distribution) in relative
entropy.
