Abstract
This paper presents a comprehensive introduction and systematic derivation of the evolutionary equations for absolute entropy H and relative entropy D, some of which exist sporadically in the literature in different forms under different subjects, within the framework of dynamical systems. In general, both H and D are dissipated, and the dissipation bears a form reminiscent of the Fisher information; in the absence of stochasticity, dH/dt is connected to the rate of phase space expansion, and D stays invariant, i.e., the separation of two probability density functions is always conserved. These formulas are validated with linear systems, and put to application with the Lorenz system and a large-dimensional stochastic quasi-geostrophic flow problem. In the Lorenz case, H falls at a constant rate with time, implying that H will eventually become negative, a situation beyond the capability of the commonly used computational technique like coarse-graining and bin counting. For the stochastic flow problem, it is first reduced to a computationally tractable low-dimensional system, using a reduced model approach, and then handled through ensemble prediction. Both the Lorenz system and the stochastic flow system are examples of self-organization in the light of uncertainty reduction. The latter particularly shows that, sometimes stochasticity may actually enhance the self-organization process.
Highlights
Having experienced the record cold in the past 20 years and a heavy, intense snow storm brought by a polar vortex during 7–9 January 2014, the City of Boston, Massachusetts, announced a state of emergency as the second polar vortex was by prediction about to approach 13 days later, with public school closures and a lot of air flight cancellations
It is interesting to note that the above formula (12) may be linked to Fisher information if the parameters, say μi, of the distribution are bound to the state variables in a form of translation such as that in a Gaussian process
This is what is shown in Cover and Thomas [25] with a Markov chain, a property that has been connected to the second law of thermodynamics
Summary
We will show that, given a dynamical system, there exist some neat laws on entropy evolution. With these laws one can in many cases avoid estimating the pdfs and overcome the above difficulties. We want to give them a comprehensive introduction and a systematic derivation within a generic framework of dynamical systems, both deterministic and stochastic As a verification, these laws are alternatively derived with linear systems in terms of means and covariances. This is followed by two applications: one with the renowned Lorenz system (Section 5), another with a quasi-geostrophic shear flow instability problem (Section 6).
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