The Peclet number of a casino: Diffusion and convection in a gambling context

The Peclet number is used to characterize the relative importance of convection over diffusion in transport phenomena. We explore an alternative yet equivalent interpretation of that classical dimensionless number in terms of the observation scale. At a microscopic scale, all phenomena are necessarily diffusive because of the randomness of molecular motion. Convection is a large-scale phenomenon, which emerges when the randomness is averaged out on a large number of microscopic events. That perspective considerably broadens the scope of the Peclet number beyond convection and diffusion: it characterizes how efficient an averaging procedure is at reducing fluctuations at a considered scale. We discuss this by drawing on a rigorous analogy with gambling: the gains and losses of an individual gambler are governed by chance, but those of a casino-the accumulated gains and losses of many gamblers-can be predicted with quasi-certainty. The Peclet number captures these scale-dependent qualitative differences.