Power-law patterns in the Phanerozoic sedimentary records of carbon, oxygen, sulfur, and strontium isotopes

Power-law patterns appear in a variety of natural systems on the modern Earth; nevertheless, whether such behaviors appeared in the deep-time environment has rarely been studied. Isotopic records in sedimentary rocks, which are widely used to reconstruct the geological/geochemical conditions in paleoenvironments and the evolutionary trajectories of biogeochemical cycles, offer an opportunity to investigate power laws in ancient geological systems. In this study, I focus on the Phanerozoic sedimentary records of carbon, oxygen, sulfur, and strontium isotopes, which have well documented and extraordinarily comprehensive datasets. I perform statistical analyses on these datasets and show that the variations in the sedimentary records of the four isotopes exhibit power-law behaviors. The exponents of these power laws range between 2.2 and 2.9; this narrow interval indicates that the variations in carbon, oxygen, sulfur, and strontium isotopes likely belong to the same universality class, suggesting that these systematic power-law patterns are governed by universal, scale-free mechanisms. I then derive a general form for these power laws from a minimalistic model based on basic physical principles and geosystem-specific assumptions, which provides an interpretation for the power-law patterns from the perspective of thermodynamics. The fundamental mechanisms regulating such patterns might have been ubiquitous in paleoenvironments, implying that similar power-law behaviors may exist in the sedimentary records of other isotopes.