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Entropy 2014, 16(7), 3605-3634; doi:10.3390/e16073605
Article

Entropy Evolution and Uncertainty Estimation with Dynamical Systems

1,2
1 School of Marine Sciences and School of Mathematics and Statistics, Nanjing University of Information Science and Technology (Nanjing Institute of Meteorology), 219 Ningliu Blvd,Nanjing 210044, China 2 China Institute for Advanced Study, Central University of Finance and Economics, 39 South College Ave, Beijing 100081, China 
Received: 4 May 2014 / Revised: 11 June 2014 / Accepted: 19 June 2014 / Published: 30 June 2014
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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.
Keywords: uncertainty estimation; entropy; second law of thermodynamics; Lorenz system; ensemble prediction; quasi-geostrophic flow; shear instability; Fisher information matrix; reduced model approach; self-organization uncertainty estimation; entropy; second law of thermodynamics; Lorenz system; ensemble prediction; quasi-geostrophic flow; shear instability; Fisher information matrix; reduced model approach; self-organization
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).
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Liang, X.S. Entropy Evolution and Uncertainty Estimation with Dynamical Systems. Entropy 2014, 16, 3605-3634.

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