Entropy: from thermodynamics to hydrology (invited talk)

In probability theory, entropy, as defined by Shannon, is none other than uncertainty quantified. The definition of entropy is very economical as it only needs the concepts of a random variable and of expectation. Commonly, probabilistic and thermodynamic entropy have been regarded as two distinct concepts having in common only the name. However, according to another school of thought, probabilistic entropy and thermodynamic entropy are logically identical concepts with only slight technical differences. Here two examples related to hydrology are used to support the latter thesis. Specifically, it is illustrated that two thermodynamic laws, the law of ideal gases and the law of phase change transition (Clausius-Clapeyron), can be derived from probabilistic entropy. The importance of the entropy concept relies on the principle of maximum entropy, which can be regarded both as a physical (ontological) principle obeyed by natural systems (cf. the Second Law of thermodynamics), as well as a logical (epistemological) principle applicable in making inference about natural systems. This principle expresses the tendency of entropy to become maximal, which constitutes the driving force of change and evolution and also offers the basis to understand and describe Nature. By maximizing entropy, i.e. uncertainty, we can describe the behaviour of physical systems. Such description is essentially probabilistic. However, if a system is composed of numerous identical elements, the uncertainty, despite being maximal at the microscopic level, at a macroscopic system it becomes as low as to yield a physical law that is in effect deterministic; for example, this is the case in the equilibrium of liquid water and water vapour. Extremal entropy considerations provide a theoretical basis also in modelling hydrological processes. However, at the high macroscopization levels related to the hydrological systems there is no hope to derive deterministic laws and thus only stochastic modelling is feasible. Linking statistical thermophysics with hydrology with a unifying view of entropy as uncertainty is a promising scientific direction.