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Entropy
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Entropy in Model Reduction
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Entropy in Model Reduction

A special issue of Entropy (ISSN 1099-4300).

Deadline for manuscript submissions: closed (28 February 2010) | Viewed by 59053

Special Issue Editor

Prof. Dr. Alexander N. Gorban

E-Mail Website
Guest Editor
Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK
Interests: neural networks; chemical and biological kinetics; human adaptation to hard living conditions; methods and technologies of collective thinking
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

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

Keywords

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

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Published Papers (6 papers)

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Research

224 KiB  
Article
Principle of Minimum Discrimination Information and Replica Dynamics
by Georgiy P. Karev
Entropy 2010, 12(7), 1673-1695; https://doi.org/10.3390/e12071673 - 28 Jun 2010
Cited by 12 | Viewed by 8119
Abstract
Dynamics of many complex systems can be described by replicator equations (RE). Here we present an effective method for solving a wide class of RE based on reduction theorems for models of inhomogeneous communities. The solutions of the RE minimize the discrimination information [...] Read more.
Dynamics of many complex systems can be described by replicator equations (RE). Here we present an effective method for solving a wide class of RE based on reduction theorems for models of inhomogeneous communities. The solutions of the RE minimize the discrimination information of the initial and current distributions at each point of the system trajectory, not only at the equilibrium, under time-dependent constraints. Applications to inhomogeneous versions of some conceptual models of mathematical biology (logistic and Ricker models of populations and Volterra’ models of communities) are given. Full article
(This article belongs to the Special Issue Entropy in Model Reduction)
539 KiB  
Article
Entropy: The Markov Ordering Approach
by Alexander N. Gorban, Pavel A. Gorban and George Judge
Entropy 2010, 12(5), 1145-1193; https://doi.org/10.3390/e12051145 - 7 May 2010
Cited by 70 | Viewed by 14391
Abstract
The focus of this article is on entropy and Markov processes. We study the properties of functionals which are invariant with respect to monotonic transformations and analyze two invariant “additivity” properties: (i) existence of a monotonic transformation which makes the functional additive with [...] Read more.
The focus of this article is on entropy and Markov processes. We study the properties of functionals which are invariant with respect to monotonic transformations and 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. All Lyapunov functionals for Markov chains which have properties (i) and (ii) are derived. We describe the most general ordering of the distribution space, with respect to which all continuous-time Markov processes are monotonic (the Markov order). The solution differs significantly from the ordering given by the inequality of entropy growth. For inference, this approach results in a convex compact set of conditionally “most random” distributions. Full article
(This article belongs to the Special Issue Entropy in Model Reduction)
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350 KiB  
Article
Engineering Model Reduction and Entropy-based Lyapunov Functions in Chemical Reaction Kinetics
by Katalin M. Hangos
Entropy 2010, 12(4), 772-797; https://doi.org/10.3390/e12040772 - 8 Apr 2010
Cited by 31 | Viewed by 9832
Abstract
In this paper, the structural properties of chemical reaction systems obeying the mass action law are investigated and related to the physical and chemical properties of the system. An entropy-based Lyapunov function candidate serves as a tool for proving structural stability, the existence [...] Read more.
In this paper, the structural properties of chemical reaction systems obeying the mass action law are investigated and related to the physical and chemical properties of the system. An entropy-based Lyapunov function candidate serves as a tool for proving structural stability, the existence of which is guaranteed by the second law of thermodynamics. The commonly used engineering model reduction methods, the so-called quasi equilibrium and quasi steady state assumption based reductions, together with the variable lumping are formally defined as model transformations acting on the reaction graph. These model reduction transformations are analysed to find conditions when (a) the reduced model remains in the same reaction kinetic system class, (b) the reduced model retains the most important properties of the original one including structural stability. It is shown that both variable lumping and quasi equilibrium based reduction preserve both the reaction kinetic form and the structural stability of reaction kinetic models of closed systems with mass action law kinetics, but this is not always the case for the reduction based on quasi steady state assumption. Full article
(This article belongs to the Special Issue Entropy in Model Reduction)
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1808 KiB  
Article
Entropy-Related Extremum Principles for Model Reduction of Dissipative Dynamical Systems
by Dirk Lebiedz
Entropy 2010, 12(4), 706-719; https://doi.org/10.3390/e12040706 - 1 Apr 2010
Cited by 21 | Viewed by 11675
Abstract
Chemical kinetic systems are modeled by dissipative ordinary differential equations involving multiple time scales. These lead to a phase flow generating anisotropic volume contraction. Kinetic model reduction methods generally exploit time scale separation into fast and slow modes, which leads to the occurrence [...] Read more.
Chemical kinetic systems are modeled by dissipative ordinary differential equations involving multiple time scales. These lead to a phase flow generating anisotropic volume contraction. Kinetic model reduction methods generally exploit time scale separation into fast and slow modes, which leads to the occurrence of low-dimensional slow invariant manifolds. The aim of this paper is to review and discuss a computational optimization approach for the numerical approximation of slow attracting manifolds based on entropy-related and geometric extremum principles for reaction trajectories. Full article
(This article belongs to the Special Issue Entropy in Model Reduction)
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455 KiB  
Article
Entropy Variation in the Two-dimensional Phase Transition of Anthracene Adsorbed at the Hg Electrode/Ethylene Glycol Solution Interface
by Claudio Fontanesi
Entropy 2010, 12(3), 570-577; https://doi.org/10.3390/e12030570 - 16 Mar 2010
Cited by 4 | Viewed by 6823
Abstract
The adsorption of anthracene (C14H10), at the mercury electrode/ethylene glycol (EG) solution interface, is characterized by a low and almost constant capacity (about 8 μF cm−2) region (capacitive “pit” or “plateau”) in capacity vs. potential curves, upon [...] Read more.
The adsorption of anthracene (C14H10), at the mercury electrode/ethylene glycol (EG) solution interface, is characterized by a low and almost constant capacity (about 8 μF cm−2) region (capacitive “pit” or “plateau”) in capacity vs. potential curves, upon selection of suitable values of temperature, bulk concentration and applied potential values. This result is rationalized assuming the occurrence of a 2D phase transition between two distinct adsorbed phases: (i) a “disordered” phase, characterized by a flat “parallel” disposition of the aromatic moiety on the electrode surface (ii) an “ordered” phase, characterized by a “perpendicular” disposition of the aromatic moiety on the electrode surface. The experimental evidence is rationalized by considering the chemical potential as an explicit function of the “electric field/adsorbed molecule” interaction. Such a modelistic approach enables the determination of the relevant standard entropy variation. Full article
(This article belongs to the Special Issue Entropy in Model Reduction)
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488 KiB  
Article
On the Entropy Based Associative Memory Model with Higher-Order Correlations
by Masahiro Nakagawa
Entropy 2010, 12(1), 136-147; https://doi.org/10.3390/e12010136 - 22 Jan 2010
Cited by 1 | Viewed by 7461
Abstract
In this paper, an entropy based associative memory model will be proposed and applied to memory retrievals with an orthogonal learning model so as to compare with the conventional model based on the quadratic Lyapunov functional to be minimized during the retrieval process. [...] Read more.
In this paper, an entropy based associative memory model will be proposed and applied to memory retrievals with an orthogonal learning model so as to compare with the conventional model based on the quadratic Lyapunov functional to be minimized during the retrieval process. In the present approach, the updating dynamics will be constructed on the basis of the entropy minimization strategy which may be reduced asymptotically to the above-mentioned conventional dynamics as a special case ignoring the higher-order correlations. According to the introduction of the entropy functional, one may involve higer-order correlation effects between neurons in a self-contained manner without any heuristic coupling coefficients as in the conventional manner. In fact we shall show such higher order coupling tensors are to be uniquely determined in the framework of the entropy based approach. From numerical results, it will be found that the presently proposed novel approach realizes much larger memory capacity than that of the quadratic Lyapunov functional approach, e.g., associatron. Full article
(This article belongs to the Special Issue Entropy in Model Reduction)
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