Macroscopic laws, microscopic dynamics, time's arrow and Boltzmann's entropy

I discuss Boltzmann's resolution of the apparent paradox: microscopic dynamics are time-symmetric but the behavior of macroscopic objects, composed of microscopic constituents, is time-asymmetric. Noting the great disparity between macroscales and microscales Boltzmann developed a statistical approach which explains the observed macroscopic behavior. In particular it predicts the increase with time of the “Boltzmann entropy”, SB(X), for “almost all” microscopic states X, of a nonequilibrium macroscopic system. The quantitative description of the macroscopic evolution, and ipso facto the compatibility between the macroscopic descriptions and microscopic descriptions, is illustrated by an example: the rigorous derivation of a diffusion equation for the typical macroscopic density profile of a Lorentz gas of independent electrons moving according to Hamiltonian dynamics. The role of low entropy “initial states” is emphasized.