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We investigate the unbiased model for money exchanges: agents give at randomtime a dollar to one another (if they have one). Surprisingly, this dynamicseventually leads to a geometric distribution of wealth (shown empirically byDragulescu and Yakovenko in [11] and rigorously in [2,12,15,18]). We prove auniform-in-time propagation of chaos result as the number of agents goes toinfinity, which links the stochastic dynamics to a deterministic infinitesystem of ordinary differential equations. This deterministic description isthen analyzed by taking advantage of several entropy-entropy dissipationinequalities and we provide a quantitative almost-exponential rate ofconvergence toward the equilibrium (geometric distribution) in relativeentropy.

From interacting agents to Boltzmann-Gibbs distribution of money