DOCSLIB.ORG

Second Order Neural Networks. Sanjoy Das Louisiana State University and Agricultural & Mechanical College

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Second Order Neural Networks. Sanjoy Das Louisiana State University and Agricultural & Mechanical College Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1994 Second Order Neural Networks. Sanjoy Das Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Das, Sanjoy, "Second Order Neural Networks." (1994). LSU Historical Dissertations and Theses. 5721. https://digitalcommons.lsu.edu/gradschool_disstheses/5721 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. University Microfilms International A Bell & Howell Information Company 300 North Zeeb Road. Ann Arbor. Ml 48106-1346 USA 313/761-4700 800/521-0600 Order Number 9502107 Second order neural networks Das, Sanjoy, Ph.D. The Louisiana State University and Agricultural and Mechanical Col., 1994 UMI 300 N. ZeebRd. Ann Arbor, MI 48106 SECOND ORDER NEURAL NETWORKS A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements of tne degree of Doctor of Philosophy in The Department of Electrical and Computer Engineering by Sanjoy Das B. S. Sambalpur University, 1985 May 1994 Acknowledgements I am grateful to Prof. S. C. Kak for his expert guidance and for the support and assistance that he has provided me for the past five years. I am also thankful to my family, especially my wife, Indira for her emotional support for the past two years, my talkative little neice, Soma for cheering me through the telephone, in times of need, and to all my friends, especially Amit, Sun- dar, Bala and Praveen for all their help. Table of Contents Acknowledgements .......................................................................................................... ii A b stra c t.............................................................................................................................. v 1. Introduction ................................................................................................................. 1 1.1 The Biological Basis of Neural Networks ................................................ 1 1.2 Artificial Neural N etw orks ........................................................................... 2 2. First Order Neural Networks .................................................................................. 6 2.1 The Gradient Descent Method ................................................................... 6 2.2 The Hopfield Neural Network .................................................................... 8 2.2.1 Dynamics of the Hopfield Neural Network .............................. 8 2.2.2 Energy of the Hopfield Neural Network ................................... 10 2.3 Other Feedback Neural Networks ............................................................... 13 2.3.1 Stochastic Feedback Neural Networks ...................................... 13 2.3.2 Lagrangian Optimization Neural Networks .............................. 15 2.3.3 Locally Connected Feedback Neural Networks ....................... 17 3. The Polynomial Neural Network .............................................................................. 19 3.1 Attractors of Dynamical System s ............................................................... 19 3.2 Computing the Roots of a Function ........................................................... 21 3.2.1 The Newton-Raphson Method for One Variable ..................... 22 3.2.2 The Multidimensional Newton-Raphson M ethod ................... 24 3.3 The Polynomial Neural Network ................................................................ 24 3.3.1 The Dynamics of the Polynomial Neural Network ............... 25 3.3.2 Computation of the Weight Matrix ............................................ 31 3.3.3 Construction of the Memory Function ..................................... 32 3.3.4 Performance of the Polynomial Neural Network .................... 37 4. Second Order Neural Network for Function Optimization ........................... 48 4.1 Locating Extremum Points of a Function ................................................. 48 4.1.1 The Newton-Raphson M ethod ..................................................... 49 4.2 Neural Networks for Optimization ............................................................. 51 4.2.1 The Basic M odel .......................................................................... 51 4.2.2 The Simplified M odel .................................................................. 54 iii 5. The G rid Connection P ro b le m ............................................................................... 74 5.1 The Grid Connection Problem ................................................................... 74 5.2 A Neural Network Solution to GCP .......................................................... 74 5.3 R esults ............................................................................................................. 76 6. C onclusions................................................................................................................... 81 6.1 D iscussions ..................................................................................................... 81 6.2 Suggestions for Further Research .............................................................. 82 Bibliography..................................................................................................................... 85 V ita ..................................................................................................................................... 89 iv Abstract In this dissertation, a feedback neural network model has been proposed. This network uses a second order method of convergence based on the Newton-Raphson method. This neural network has both discrete as well as continuous versions. When used as an associative memory, the proposed model has been called the polynomial neural network (PNN). The memories of this network can be located anywhere in an n dimensional space rather than being confined to the comers of the latter. A method for storing memories has been proposed. This is a single step method unlike the currently known computationally intensive iterative methods. An energy function for the polynomial neural network has been suggested. Issues relating to the error- correcting ability of this network have been addressed. Additionally, it has been found that the attractor basins of the memories of this network reveal a curious frac­ tal topology, thereby suggesting a highly complex and often unpredictable nature. The use of the second order neural network as a function optimizer has also been shown. While issues relating to the hardware realization of this network have only been addressed briefly, it has been indicated that such a network would have a large amount of hardware for its realization. This problem can be obviated by using a simplified model that has also been described. The performance of this simplified model is comparable to that of the basic model while requiring much less hardware for its realization. v Chapter 1 Introduction 1.1 Biological Basis of Neural Networks The brain is one the most fascinating structures in nature. It is the seat of memory, emotion and learning. It is also the organ from which all intelligent behavior emerges. The brain is a vastly complicated structure consisting of numerous cells that are called neurons. Investigations have shown that these neurons are essentially two-state cells that can be either firing or quiescent. The neurons are connected to one another by means of dendrites, that emerge from each of these neurons. The neuronal connections themselves are referred to as synapses, and it is by means of these synapses that neu­ rons exert excitatory or inhibitory influences on one another. The number of neurons in the human brain is estimated to be of the order of 1010, with the number of neuron­
Load more

Recommended publications
  • Dissipative Mechanics Using Complex-Valued Hamiltonians
    Dissipative Mechanics Using Complex-Valued Hamiltonians S. G. Rajeev Department of Physics and Astronomy University of Rochester, Rochester, New York 14627 March 18, 2018 Abstract We show that a large class of dissipative systems can be brought to a canonical form by introducing complex co-ordinates in phase space and a complex-valued hamiltonian. A naive canonical quantization of these sys- tems lead to non-hermitean hamiltonian operators. The excited states are arXiv:quant-ph/0701141v1 19 Jan 2007 unstable and decay to the ground state . We also compute the tunneling amplitude across a potential barrier. 1 1 Introduction In many physical situations, loss of energy of the system under study to the out- side environment cannot be ignored. Often, the long time behavior of the system is determined by this loss of energy, leading to interesting phenomena such as attractors. There is an extensive literature on dissipative systems at both the classical and quantum levels (See for example the textbooks [1, 2, 3]). Often the theory is based on an evolution equation of the density matrix of a ‘small system’ coupled to a ‘reservoir’ with a large number of degrees of freedom, after the reservoir has been averaged out. In such approaches the system is described by a mixed state rather than a pure state: in quantum mechanics by a density instead of a wavefunction and in classical mechanics by a density function rather than a point in the phase space. There are other approaches that do deal with the evolution equations of a pure state. The canonical formulation of classical mechanics does not apply in a direct way to dissipative systems because the hamiltonian usually has the meaning of energy and would be conserved.
    [Show full text]
  • An Ab Initio Definition of Life Pertaining to Astrobiology Ian Von Hegner
    An ab initio definition of life pertaining to Astrobiology Ian von Hegner To cite this version: Ian von Hegner. An ab initio definition of life pertaining to Astrobiology. 2019. hal-02272413v2 HAL Id: hal-02272413 https://hal.archives-ouvertes.fr/hal-02272413v2 Preprint submitted on 15 Oct 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. HAL archives-ouvertes.fr | CCSD, August, 2019. An ab initio definition of life pertaining to Astrobiology Ian von Hegner Aarhus University Abstract Many definitions of life have been put forward in the course of time, but none have emerged to entirely encapsulate life. Putting forward an adequate definition is not a simple matter to do, despite many people seeming to believe they have an intuitive understanding of what is meant when it is stated that something is life. However, it is important to define life, because we ourselves, individually and collectively, are life, which entails an importance in itself. Furthermore, humankind‘s capability to look for life on other planets is steadily becoming a real possibility. But in order to realize that search, a definition of life is required.
    [Show full text]
  • Dissipative Structures in Educational Change: Prigogine and the Academy
    Wichita State University Libraries SOAR: Shocker Open Access Repository Donald L. Gilstrap University Libraries Dissipative Structures in Educational Change: Prigogine and the Academy Donald L. Gilstrap Wichita State University Citation Gilstrap, Donald L. 2007. Dissipative structures in educational change: Prigogine and the academy. -- International Journal of Leadership in Education: Theory and Practice; v.10 no.1, pp. 49-69. A post-print of this paper is posted in the Shocker Open Access Repository: http://soar.wichita.edu/handle/10057/6944 Dissipative structures in educational change: Prigogine and the academy Donald L. Gilstrap This article is an interpretive study of the theory of irreversible and dissipative systems process transformation of Nobel Prize winning physicist Ilya Prigogine and how it relates to the phenomenological study of leadership and organizational change in educational settings. Background analysis on the works of Prigogine is included as a foundation for human inquiry and metaphor generation for open, dissipative systems in educational settings. International case study research by contemporary systems and leadership theorists on dissipative structures theory has also been included to form the interpretive framework for exploring alternative models of leadership theory in far from equilibrium educational settings. Interpretive analysis explores the metaphorical significance, connectedness, and inference of dissipative systems and helps further our knowledge of human-centered transformations in schools and colleges. Introduction Over the course of the past 30 years, educational institutions have undergone dramatic shifts, leading teachers and administrators in new and sometimes competing directions. We are all too familiar with rising enrolments and continual setbacks in financial resources. Perhaps this has never been more evident, particularly in the USA, than with the recent changes in educational and economic policies.
    [Show full text]
  • Entropy and Life
    Entropy and life Research concerning the relationship between the thermodynamics textbook, states, after speaking about thermodynamic quantity entropy and the evolution of the laws of the physical world, that “there are none that life began around the turn of the 20th century. In 1910, are established on a firmer basis than the two general American historian Henry Adams printed and distributed propositions of Joule and Carnot; which constitute the to university libraries and history professors the small vol- fundamental laws of our subject.” McCulloch then goes ume A Letter to American Teachers of History proposing on to show that these two laws may be combined in a sin- a theory of history based on the second law of thermo- gle expression as follows: dynamics and on the principle of entropy.[1][2] The 1944 book What is Life? by Nobel-laureate physicist Erwin Schrödinger stimulated research in the field. In his book, Z dQ Schrödinger originally stated that life feeds on negative S = entropy, or negentropy as it is sometimes called, but in a τ later edition corrected himself in response to complaints and stated the true source is free energy. More recent where work has restricted the discussion to Gibbs free energy because biological processes on Earth normally occur at S = entropy a constant temperature and pressure, such as in the atmo- dQ = a differential amount of heat sphere or at the bottom of an ocean, but not across both passed into a thermodynamic sys- over short periods of time for individual organisms. tem τ = absolute temperature 1 Origin McCulloch then declares that the applications of these two laws, i.e.
    [Show full text]
  • Role of Nonlinear Dynamics and Chaos in Applied Sciences
    v.;.;.:.:.:.;.;.^ ROLE OF NONLINEAR DYNAMICS AND CHAOS IN APPLIED SCIENCES by Quissan V. Lawande and Nirupam Maiti Theoretical Physics Oivisipn 2000 Please be aware that all of the Missing Pages in this document were originally blank pages BARC/2OOO/E/OO3 GOVERNMENT OF INDIA ATOMIC ENERGY COMMISSION ROLE OF NONLINEAR DYNAMICS AND CHAOS IN APPLIED SCIENCES by Quissan V. Lawande and Nirupam Maiti Theoretical Physics Division BHABHA ATOMIC RESEARCH CENTRE MUMBAI, INDIA 2000 BARC/2000/E/003 BIBLIOGRAPHIC DESCRIPTION SHEET FOR TECHNICAL REPORT (as per IS : 9400 - 1980) 01 Security classification: Unclassified • 02 Distribution: External 03 Report status: New 04 Series: BARC External • 05 Report type: Technical Report 06 Report No. : BARC/2000/E/003 07 Part No. or Volume No. : 08 Contract No.: 10 Title and subtitle: Role of nonlinear dynamics and chaos in applied sciences 11 Collation: 111 p., figs., ills. 13 Project No. : 20 Personal authors): Quissan V. Lawande; Nirupam Maiti 21 Affiliation ofauthor(s): Theoretical Physics Division, Bhabha Atomic Research Centre, Mumbai 22 Corporate authoifs): Bhabha Atomic Research Centre, Mumbai - 400 085 23 Originating unit : Theoretical Physics Division, BARC, Mumbai 24 Sponsors) Name: Department of Atomic Energy Type: Government Contd...(ii) -l- 30 Date of submission: January 2000 31 Publication/Issue date: February 2000 40 Publisher/Distributor: Head, Library and Information Services Division, Bhabha Atomic Research Centre, Mumbai 42 Form of distribution: Hard copy 50 Language of text: English 51 Language of summary: English 52 No. of references: 40 refs. 53 Gives data on: Abstract: Nonlinear dynamics manifests itself in a number of phenomena in both laboratory and day to day dealings.
    [Show full text]
  • Molecules, Information and the Origin of Life: What Is Next?
    molecules Perspective Molecules, Information and the Origin of Life: What Is Next? Salvatore Chirumbolo 1,* and Antonio Vella 2 1 Department of Neurosciences, Biomedicine and Movement Sciences, University of Verona, 37134 Verona, Italy 2 Verona-Unit of Immunology, Azienda Ospedaliera Universitaria Integrata, 37134 Verona, Italy; [email protected] * Correspondence: [email protected]; Tel.: +39-0458027645 Abstract: How life did originate and what is life, in its deepest foundation? The texture of life is known to be held by molecules and their chemical-physical laws, yet a thorough elucidation of the aforementioned questions still stands as a puzzling challenge for science. Focusing solely on molecules and their laws has indirectly consolidated, in the scientific knowledge, a mechanistic (reductionist) perspective of biology and medicine. This occurred throughout the long historical path of experimental science, affecting subsequently the onset of the many theses and speculations about the origin of life and its maintenance. Actually, defining what is life, asks for a novel epistemology, a ground on which living systems’ organization, whose origin is still questioned via chemistry, physics and even philosophy, may provide a new key to focus onto the complex nature of the human being. In this scenario, many issues, such as the role of information and water structure, have been long time neglected from the theoretical basis on the origin of life and marginalized as a kind of scenic backstage. On the contrary, applied science and technology went ahead on considering molecules as the sole leading components in the scenery. Water physics and information dynamics may have a role in living systems much more fundamental than ever expected.
    [Show full text]
  • Ordered Complexity from Dissipative and Chaotic Systems, Including the Human Brain and Society and the Universe; Relevance of the Second Law of Thermodynamics
    Volume 6, Issue 8, August – 2021 International Journal of Innovative Science and Research Technology ISSN No:-2456-2165 Ordered Complexity from Dissipative and Chaotic Systems, Including the human brain and society and the Universe; Relevance of the Second Law of Thermodynamics Authors Sarbajit Roy, Shailendra Kumar Pandey, Vishal Kumar, Purushotam Kumar, Sudhir Kumar, Raman Kumar Mandal, Rajnish Kumar, Deepshikha Mishra. RamGovind Institute of Technology, Koderma, Jharkhand Abstract:- The paper connects the characteristics of a A living system is characterized by continual flow and dissipative system, which operates far from equilibrium, change in its equilibrium, a process which requires a chaotic system , depending upon initial conditions and thousands of chemical reactions. When these processes complexity, which forms from the holistic connection of come to a standstill, they induce chemical and thermal its parts .Consequent, the human brain, human society equilibrium, which also signifies a dead organism. Living and the universe itself are portrayed as dissipative, organisms continually maintain themselves in a state far chaotic and complex. Chaos is everywhere as is from equilibrium, which is a state of life. complexity. The paper analyzes the aspects of nonlinearity in system dynamics based on the ideas of Although far from equilibrium , the state is stable and Ilya Prigogine and attempts to forge a link between the over long periods of time. In spite of the ongoing flow of systems discussed . The Second Law of Thermodynamics components, the same structure is maintained . involving the total increase in entropy is elucidated as being universal and inherent in all events of the universe. To describe systems far from equilibrium, Prigogine The law may be considered as part of the evolutionary realized that classical thermodynamics was inappropriate, tool in the formation of complex organisms, galaxies and since its mathematical structure was linear.
    [Show full text]
  • Thermodynamics and Life: an Evolutionary Point of View
    N. Marchettini et al., Int. J. of Design & Nature and Ecodynamics. Vol. 3, No. 4 (2009) 249–258 THERMODYNAMICS AND LIFE: AN EVOLUTIONARY POINT OF VIEW N. MARCHETTINI1, E. SIMONCINI1 & E.B.P. TIEZZI2 1Department of Chemistry, University of Siena, Siena, Italy. 2Department of Mathematics and Informatics, University of Siena, Siena, Italy. ABSTRACT The complexity we observe in life forms, in their development and in biological evolution, is the result of the temporal and spatial constraints of a long evolutionary history made up of relations accumulating in time. This work assumes that space and time belong to different logical types: the fi rst being reversible and conservative and the second irreversible and evolutionary. Time and space are reciprocally irreducible quantities. We refer to the thermodynamics of the Nobel laureate in Chemistry, Ilya Prigogine, to perform a new point of view of life, in terms of evolutionary thermodynamics. Keywords: Dissipative Structures, Evolution, Information, life origins, Macrostates, Microstates,Tempos. 1 INTRODUCTION ON PRIGOGINE RESEARCH The historical–scientifi c moment that we are living in seems to be searching for, once again, a defi nition for a concept that has had complex consequences. The concept we will be talking about in this work is that of dissipative structures, as its ‘inventor’, Ilya Prigogine, has named it. The magnitude of Prigogine theory can be understood only if time is considered in its closed relation with life and evolution [1–4]. The comprehension of irreversible phenomena has undergone a revolution, thanks to the study of dissipative structures, of those systems, that is, far from equilibrium and open to the exterior, in which fl uctuations of energy can produce order out of chaos.
    [Show full text]
  • The Entropocene
    THE ENTROPOCENE Jeff Robbins Rutgers University 101 Murray Hall, New Brunswick, NJ 08901, USA ABSTRACT According to Eric Schneider and the late James Kay “nature abhors a gradient” as it seeks to degrade any and all differences that can make a difference. They also claim that life self-organized by creating a meta-order out of disordered orders, evolving as an increasingly efficient, effective, sustainable (order from order) means for degrading the huge sun to earth temperature gradient. They inject purpose into the scheme by claiming that life represents “order emerging from disorder in the service of more disorder.” Or, as Jeffrey Wicken succinctly put it in “Evolution and Thermodynamics: The New Paradigm,” “Organisms are remote-from-equilibrium systems that maintain their organizational structures by irreversibly degrading free energy through informed kinetic pathways acquired through evolution. Dissipation through structuring is the strategy of life.” In other words, life came about as an evolving means for giving the Second Law of Thermodynamics what it wants. Entropy. In my 2015 ISSS paper, “Anthropocene as Life’s State of the Art in Disorder Production: A Sustainability Conundrum,” I proposed that our species collectively is disorder producer summa cum laude. We are the most efficient, most effective, degrader of gradients, producer of entropy yet evolved. Calling our epoch, “the Anthropocene” doesn’t capture our essence. Our epoch is the Entropocene. We have turned the bio-geosphere from an accumulator of solar exergy (“free energy”, the measure of energy’s utility) as in fossil fuels, into a trapper of entropy a.k.a., global warming.
    [Show full text]
  • Introduction to the Theory of Infinite-Dimensional Dissipative Systems
    I. D. Chueshov Author: I. D. Chueshov This book provides an exhau - stive introduction to the scope Introduction to the Theory Title: Introduction to the Theory of InfiniteDimensional of InfiniteDimensional of main ideas and methods of the Dissipative Systems Dissipative Systems theory of infinite-dimensional dis - sipative dynamical systems which 966–7021–64–5 ISBN: 966–7021–64–5 has been rapidly developing in re - cent years. In the examples sys tems generated by nonlinear partial differential equations I.D.Chueshov I . D . Chueshov arising in the different problems of modern mechanics of continua are considered. The main goal of the book is to help the reader Introduction Theory Introduction to the Theory to master the basic strategies used Infinite-Dimensional of Infinite-Dimensional in the study of infinite-dimensional dissipative systems and to qualify Dissipative Dissipative him/her for an independent scien - Systems 2002 Systems tific research in the given branch. Experts in nonlinear dynamics will find many fundamental facts in the convenient and practical form » in this book. Universitylecturesincontemporarymathematics The core of the book is com - posed of the courses given by the author at the Department of Me chanics and Mathematics at Kharkov University during a number of years. This book con - tains a large number of exercises which make the main text more complete. It is sufficient to know the fundamentals of functional analysis and ordinary differential equations to read the book. CTA Translated by Constantin I. Chueshov ORDER You can ORDER this book from the Russian edition («ACTA», 1999) while visiting the website of «ACTA» Scientific Publishing House Translation edited by www.acta.com.ua http://www.acta.com.uawww.acta.com.ua/en/ Maryna B.
    [Show full text]
  • Biological Cybernetics C Springer-Verlag 1996
    Biol. Cybern. 75, 229–238 (1996) Biological Cybernetics c Springer-Verlag 1996 A complex-valued associative memory for storing patterns as oscillatory states Srinivasa V. Chakravarthy, Joydeep Ghosh Department of Electrical and Computer Engineering, The University of Texas, Austin, TX 78712, USA Received: 4 December 1995 / Accepted in revised form: 18 June 1996 Abstract. A neuron model in which the neuron state is de- Real neural systems exhibit a range of phenomena such scribed by a complex number is proposed. A network of as oscillations, frequency entrainment, phase-locking, and these neurons, which can be used as an associative memory, even chaos, which are rarely touched by applied models. operates in two distinct modes: (i) fixed point mode and In particular, a great range of biological neural behavior (ii) oscillatory mode. Mode selection can be done by vary- cannot be modeled by networks with only fixed point be- ing a continuous mode parameter, ν, between 0 and 1. At havior. There is strong evidence that brains do not store one extreme value of ν (= 0), the network has conservative patterns as fixed points. For example, work done in the dynamics, and at the other (ν = 1), the dynamics are dissi- last decade by Freeman and his group with mammalian ol- pative and governed by a Lyapunov function. Patterns can factory cortex revealed that odors are stored as oscillating be stored and retrieved at any value of ν by, (i) a one-step states (Skarda and Freeman 1987). Deppisch et al. (1994) outer product rule or (ii) adaptive Hebbian learning. In the modeled synchronization, switching and bursting activity fixed point mode patterns are stored as fixed points, whereas in cat visual cortex using a network of spiking neurons.
    [Show full text]
  • Calculation of the Entropy Balance Equation in a Non-Equilibrium Reaction-Diffusion System
    Entropy 2010, 12, 2436-2449; doi:10.3390/e12122436 OPEN ACCESS entropy ISSN 1099-4300 www.mdpi.com/journal/entropy Review Calculation of the Entropy Balance Equation in a Non-equilibrium Reaction-diffusion System Hitoshi Mahara * and Tomohiko Yamaguchi Nanosystems Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Higashi, Tsukuba, Ibaraki, 305-8565, Japan; E-Mail: [email protected] * Author to whom correspondence should be addressed; E-Mail: [email protected]. Received: 5 November 2010; in revised form: 10 December 2010 / Accepted: 11 December 2010 / Published: 17 December 2010 Abstract: In this article, we discuss the relationships between thermodynamic quantities and the spatial pattern in a reaction-diffusion model based on the reversible Gray-Scott model. This model is introduced for calculation of the entropy production in a reaction-diffusion system. First, we show the relationship between entropy production and pattern formation, and suggest that the entropy production could be an index of different patterns. Then the entropy production is applied for searching the parameter region where the pattern is bistable. Moreover, the entropy change is calculated by using the relative chemical potential that is defined based on the equilibrium state and not on the standard chemical potential. The results of the entropy change are consistent with the intrinsic property of the entropy, therefore, the entropy change calculated in this way may be regarded as an appropriate quantity for the discussion of the thermodynamic properties in a non equilibrium system. Keywords: entropy balance; reversible Gray-Scott model; reaction-diffusion system; entropy production; entropy change; entropy flow; entropy balance equation PACS Codes:82.40.Ck, 05.65.+b, 05.70.-a, 82.20.Wt Entropy 2010, 12 2437 1.
    [Show full text]