A novel stochastic foundation of entropy and its applicability to hydrometeorology

In the framework of the books Hydroclimatic Stochastics and Stochastics as Physics, a novel stochastic foundation of entropy is introduced. Entropy is treated as a purely probabilistic concept — a rigorous quantification of uncertainty — derived entirely within probability theory and stochastic processes, without deterministic or metaphorical interpretations. It emerges naturally from the principle of maximum entropy, unifying thermodynamic origins with statistical descriptions of complex systems. This approach allows key properties in atmospheric thermodynamics to be derived deductively, without presupposing thermodynamic principles. In hydrometeorology, it enables parsimonious modelling of rainfall, temperature, and other processes exhibiting scaling, persistence, and intermittency. It further supports robust inference under uncertainty (by induction), improved stochastic simulation of extremes, and a deeper understanding of long-term variability and climate dynamics through entropy extremization constrained by conservation laws. Applications illustrate the linkage between micro-scale stochastic behaviour and macro-scale hydrometeorological responses, offering a coherent physical-stochastic paradigm for geophysical systems.