Abstract
Some known results from statistical thermophysics as well as from hydrology are revisited from a different perspective trying: (a) to unify the notion of entropy in thermodynamic and statistical/stochastic approaches of complex hydrological systems and (b) to show the power of entropy and the principle of maximum entropy in inference, both deductive and inductive. The capability for deductive reasoning is illustrated by deriving the law of phase change transition of water (Clausius-Clapeyron) from scratch by maximizing entropy in a formal probabilistic frame. However, such deductive reasoning cannot work in more complex hydrological systems with diverse elements, yet the entropy maximization framework can help in inductive inference, necessarily based on data. Several examples of this type are provided in an attempt to link statistical thermophysics with hydrology with a unifying view of entropy.
Highlights
Uncertainty is the only certainty there is, and knowing how to live with insecurity is the only security.(Paulos [1], quoting his father)Entropy is etymologized from the ancient Greek ἐντροπία but was introduced as a scientific term by Rudolf Clausius only in 1865, the concept appears in his earlier works
The law is very old (Clausius-Clapeyron), the derivation provided is new and constitutes an ideal example of how by maximizing entropy, i.e., uncertainty, at the microscopic level we can derive a physical law which virtually expresses certainty at a macroscopic level. Such deductive reasoning cannot work in more complex hydrological systems with diverse elements, yet the entropy maximization framework can help in inductive inference, necessarily based on data
Koutsoyiannis [9] suggested the use of entropy production in logarithmic time (EPLT) in a continuous time representation of the process of interest
Summary
Uncertainty is the only certainty there is, and knowing how to live with insecurity is the only security. The law is very old (Clausius-Clapeyron), the derivation provided is new and constitutes an ideal example of how by maximizing entropy, i.e., uncertainty, at the microscopic level we can derive a physical law which virtually expresses certainty at a macroscopic level Such deductive reasoning cannot work in more complex hydrological systems with diverse elements, yet the entropy maximization framework can help in inductive inference, necessarily based on data. The paper is written as a self-contained study with a pedagogic style in an attempt to help students of the entropy topic to avoid some confusion which all of us may have experienced For this reason, the paper contains a logical and mathematical foundation of the concepts used (Section 2) as well as several side derivations necessary to make the presentation complete (Section 3 and Appendix)
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