  Continuous-time random walks offer powerful coarse-grained descriptions of
transport processes. We here microscopically derive such a model for a Brownian
particle diffusing in a deep periodic potential. We determine both the
waiting-time and the jump-length distributions in terms of the parameters of
the system, from which we analytically deduce the non-Gaussian characteristic
function. We apply this continuous-time random walk model to characterize the
underdamped diffusion of single Cesium atoms in a one-dimensional optical
lattice. We observe excellent agreement between experimental and theoretical
characteristic functions, without any free parameter.
