On the derivation of hydrodynamics from the Boltzmann equation

We review the main ideas on the derivation of hydrodynamical equations from microscopic models. The Boltzmann equation, which is a good approximation for the evolution of rare gases, provides a useful tool to test these ideas in mathematically controllable situations such as the Euler and incompress ible Navier-Stokes limits, which we describe in some detail. We also discus s the heuristics and some rigorous results available for stochastic particl e systems. (C) 1999 American Institute of Physics. [S1070-6631(99)02008-5].