Special Issue "Entropy in Model Reduction"

Guest Editor
Prof. Dr. Alexander Gorban
Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK
Website: http://www.math.le.ac.uk/people/ag153/
E-Mail:
Interests: neural networks; chemical and biological kinetics; human adaptation to hard living conditions; methods and technologies of collective thinking

Dear Colleagues,

In the practice of modeling of complex systems tools are necessary for the construction of models which have appropriate complexity, accuracy and do not violate the basic laws. Methods based on entropy allow us to reduce the model’s complexity. At the same time, entropy based methods give the possibility to produce models which follow some basic principles: do not produce information from nothing, satisfy the second law of thermodynamics and have other attractive properties. The area of applications of these methods is enormously wide: from physics and chemistry to biology, psychology, sociology and economics.

In this volume we invite papers which propose, review and analyse entropic methods for model reduction and for analysis of reduced models. Works with various applications of entropic methods for model reduction are also welcome.

Prof. Dr. Alexander Gorban
Guest Editor

Submission

All manuscripts should be submitted to entropy@mdpi.org with a copy to the Guest Editor. Manuscripts can be submitted until the deadline. Papers will be published continuously (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are refereed through a peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed Open Access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this Open Access journal is 1000 CHF per accepted paper.

• model reduction
• complexity reduction
• dissipativity preservation
• physical kinetics
• chemical kinetics
• systems biology
• econophysics
• sociophysics

Title: Replicator dynamics and the principle of maximum relative entropy
Author: Georgy Karev
Affiliation: Lockheed Martin MSD, National Institute of Health, Bethesda, MD 20894, USA
Abstract: Dynamics of many complex systems in different areas can be described by the replicator equations (RE). In mathematical biology RE describe the distributions of inhomogeneous populations and communities. We show that solutions of a wide class of replicator equations minimize the relative entropy of the distribution at each point of the system trajectory, not only at the equilibrium, under time-dependent constraints, which, in their turn, can be computed explicitly at every instant due to the system dynamics. Therefore, the maximum relative entropy principle, for systems governed by the replicator equations can be derived from the system dynamics rather than postulated. Applications to the inhomogeneous models of populations and communities are given

Type of Paper: Article
Title: Engineering Model Reduction and Entropy-Based Lyapunov Functions in Chemical Reaction Kinetics
Author: Katalin M. Hangos
Affiliation: Process Control Research Group, Computer and Automation Research Institute of HAS, Budapest, Hungary; E-Mail: hangos@daedalus.scl.sztaki.hu
Abstract: In this paper the original notion of chemical reaction systems obeying the mass action law is generalized to cover the cases with real exponents in the reaction monomials such, that the generalized class still possesses the same stability property under the usual entropy-based Lyapunov function. The commonly used engineering model reduction methods, the so-called steady-state assumption based reduction and the variable lumping are then formally defined and analyzed to find conditions when (a) the reduced model remains in the same generalized reaction kinetic class, (b) the reduced model retains the most important dynamic properties of the original one: its controllability, observability and stability. The methods will be illustrated on simple bio-chemical reaction system examples.

Type of Paper: Article
Title: Prediction of the Suitable Coverage for Vegetation Restoration in Arid Area of China
Author: Yuanrun Zheng
Affiliation: Institute of Botany, Chinese Academy of Sciences, No. 20 Nanxincun, Xiangshan, Beijing 100093, China; E-Mail: zhengyr@ibcas.ac.cn
Abstract: It appears that two main factors hinder the effective incorporation of ecological information out of computer modeling into resource management at large spatial scales: (1) some models are not ecologically sound; (2) most models based on justified ecological principles are often too sophisticated or detailed. This paper used a generic model based on well established ecological principles, while with appropriate details to better server sustainable resource management decision making.
The model was used to simulate the annual growth, Foliage Projective Cover (FPC), and evaporative coefficient in large arid areas in northwest China, where serious desertification happened recently. Observed NPP data in the study area were used to validate the model, and the model results were in high agreement with observed data. Then the model was used to simulate evaporative coefficient, FPC, and annual production in the study area. The simulation results indicate that except several sites, the k parameter is lower than 0.35×10^-2 and implies a typical arid climate in study area. Estimated foliage projective cover of plant community is lower than 50% in most sites, and the annual production is very low -- less than 1 tonnes^.ha^-1.yr^-1in 93.8% of sites. These simulation results could serve as good references for vegetation restoration and livestock husbandry management in arid areas in China.

Type of Paper: Article
Title: Entropy: The Markov Ordering Approach
Authors: Alexander N. Gorban ¹, Pavel A. Gorban ² and George Judge ³
Affiliations: ¹ Department of Mathematics, University of Leicester, UK; E-Mail: ag153@le.ac.uk
² Siberian Federal University, Krasnoyarsk, Russia; E-Mail: pavelgorban@yandex.ru
³ Department of Resource Economics, University of California, Berkeley, CA, USA; E-Mail: gjudge@berkeley.edu
Abstract: The focus of this article is on entropy and Markov processes. Classical entropy increases in all Markov processes with uniform equilibrium distributions. This is why we may say that the distribution with higher entropy is more random, and why we use entropy conditional maximization for the evaluation of the probability distribution when our information is partial and incomplete. Other functionals exist, which increase in all Markov processes. We describe the most general ordering of the distribution space, with respect to which all Markov processes are monotonic (in continuous time), in the Markov order. The solution differs significantly from the ordering given by the inequality of entropy growth: Q is more random than P if S(P) < S(Q). This approach results in a convex compact set of conditionally "most random" distributions, which includes the conditional entropy maximizers of all Lyapunov functionals.
A monotonic function of a Lyapunov functional also changes monotonically in time and we study the properties of functional which are invariant with respect to monotonic transformations. We analyze two invariant "additivity" properties: (i) existence of a monotonic transformation which makes the functional additive with respect to the joining of independent systems and (ii) existence of a monotonic transformation which makes the functional additive with respect to the partitioning of the space of states. We derived all Lyapunov functionals for Markov chains which have properties (i) and (ii). They form two, one-parametric families: one corresponds to the Cressie-Read family, the other is a convex combination of the classical Boltzmann--Gibbs--Shannon entropy and the Burg entropy.