Understanding why and how biological diversity changes through geologic time is one of the most fundamental yet challenging goals of paleobiology. Using the Paleobiology Database and Sepkoski’s compendium we demonstrate that complex patterns in the distribution of origination and extinction events are the result of a simple underlying process analogous to non-equilibrium dynamics in statistical mechanics. We show that fluctuations in marine biodiversity over the past 550 million years result from the superposition of many independently diversifying subsystems whose fluctuations are Gaussian. These independent subsystems correspond to lineages of closely related animal taxa, indicating that diversification within lineages is driven by random additive interactions with the environment. We also demonstrate that the simple interactions between lineages and their environment change slowly through time, most likely the result of non-random evolutionary changes in the physiology and demography of new lineages. This variation leads to patterns of apparent complexity when lineages are analyzed in combination. Our theory sheds light on how seemingly complicated biodiversity patterns in the fossil record share common mechanisms with other striking features of our world such as non-equilibrium physical systems (e.g. climate), and vagaries of the stock market.