Recently Cahn's generalized diffusion equation theory of spinodal decomposition in binary alloys has been modified to include the effects of thermal fluctuations. This paper reports studies of a one-dimensional binary alloy system in which fluctuations can be observed on an atomic time scale. The system, a computer-simulated linear chain binary alloy which evolves from an initially random atomic arrangement through interchange of unlike nearest neighbors via the Monte Carlo technique, rapidly develops grains of two different concentrations and then slowly experiences coarsening. A numerical solution of the diffusion equation successfully predicts the development of grain structure, but only predicts coarsening to the extent present as fluctuations in the initial atomic arrangement. The simulated alloy coarsens further than the prediction of the diffusion equation because of thermal fluctuations which develop naturally during its evolution. This suggests that thermal fluctuations may play an important role in coarsening in real alloys.