The settling of a Brownian particle in a semi-infinite fluid bounded by a bottom plane is studied on the basis of Smoluchowski’s exact solution of the equation describing diffusion in the gravitational potential. Expressions are derived for the mean height and the variance of height at some time after starting at an initial height. These quantities show interesting behavior as a function of time. It is shown that for certain initial heights the Boltzmann entropy does not increase steadily. It increases at first but then decreases to its equilibrium value.