Entropy Evolution and Uncertainty Estimation with Dynamical Systems

Entropy

关键词: uncertainty estimation;    entropy;    second law of thermodynamics;    Lorenz system;    ensemble prediction;    quasi-geostrophic flow;    shear instability;    Fisher information matrix;    reduced model approach;    self-organization;   

DOI  :  10.3390/e16073605

来源: mdpi

【 摘 要 】

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.

【 授权许可】

CC BY   
© 2014 by the authors; licensee MDPI, Basel, Switzerland