Abstract
While the modern definition of entropy is genuinely probabilistic, in entropy production the classical thermodynamic definition, as in heat transfer, is typically used. Here we explore the concept of entropy production within stochastics and, particularly, two forms of entropy production in logarithmic time, unconditionally (EPLT) or conditionally on the past and present having been observed (CEPLT). We study the theoretical properties of both forms, in general and in application to a broad set of stochastic processes. A main question investigated, related to model identification and fitting from data, is how to estimate the entropy production from a time series. It turns out that there is a link of the EPLT with the climacogram, and of the CEPLT with two additional tools introduced here, namely the differenced climacogram and the climacospectrum. In particular, EPLT and CEPLT are related to slopes of log-log plots of these tools, with the asymptotic slopes at the tails being most important as they justify the emergence of scaling laws of second-order characteristics of stochastic processes. As a real-world application, we use an extraordinary long time series of turbulent velocity and show how a parsimonious stochastic model can be identified and fitted using the tools developed.
Highlights
Entropy was first recognized as a probabilistic concept in 1887 by Boltzmann [1], who established a relationship of entropy with probabilities of statistical mechanical system states, explaining the Second Law of Thermodynamics as the tendency of the system to run toward more probable states
A decade later, probabilistic entropy and the principle of maximum entropy were used in geophysical sciences and hydrology, initially for parameter estimation of models [4] and probability distributions [5]
In the previous subsection we have shown that the EPLT and the CEPLT are related to log-log derivative (LLD) or slopes of log-log plots of second order tools such as climacogram, climacospectrum, power spectrum, etc
Summary
Entropy was first recognized as a probabilistic concept in 1887 by Boltzmann [1], who established a relationship of entropy with probabilities of statistical mechanical system states, explaining the Second Law of Thermodynamics as the tendency of the system to run toward more probable states. Niven and Ozawa [19] provide a general definition of entropy production along with a brief review of applications of extremum principles in geophysics and hydrology. By analogy to classical thermodynamic versions, we may define entropy production in stochastics as the derivative of (probabilistic) entropy with respect to time. In an earlier study [22], the derivative with respect to the logarithm of time was introduced and termed ‘entropy production in logarithmic time’ (EPLT). We advance the study of the EPLT concept, with particular emphasis on the conditional entropy production (CEPLT), when the past and present have been observed (Section 2.2). The theoretical properties of EPLT and CEPLT are studied in a general setting and in application to a broad set of specific types of stochastic processes (Sections 3.1 and 3.2). While the two appendices are made in a stand-alone form separate from the body of the article, they are perhaps the most essential part of the study
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