Intro to Complex Systems Up to Ch 6 PDF

Paul Fieguth An Introduction to Complex Systems Society, Ecology, and Nonlinear Dynamics Second Edition An Introduction to Complex Systems Paul Fieguth An Introduction to Complex Systems Society, Ecology, and Nonlinear Dynamics Second Edition 123 Paul Fieguth Faculty of Engineering University of Waterloo Waterloo, ON, Canada ISBN 978-3-030-63167-3 ISBN 978-3-030-63168-0 (eBook) https://doi.org/10.1007/978-3-030-63168-0 1st edition: © Springer International Publishing Switzerland 2017 2nd edition: © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface to the Second Edition (2020) In the years since the ﬁrst edition of this textbook appeared, I have had the good fortune to teach complex systems to a new set of undergraduate students every winter. It is such students who have motivated my passion for the topic, and who have challenged me to continually reﬁne the clarity of my teaching. In addition, the practical aspects of teaching a course year after year has led to corrected errors, additional examples, illustrations, and end of chapter questions to improve the depth and breadth of student learning. The topics in this text are as relevant as ever. Probably the biggest impetus for a new edition was the Covid-19 pandemic, which burst onto the international scene in 2020, and which has complex systems written all over it. In media discussions of Covid-19, I routinely encountered misunderstandings of exponentials (Example 5.1), rare events/power-laws (Chapter 9), and nonlinear dynamics (Chapter 7), so the pandemic felt like an opportune time (and indeed presented a compelling need) to re-examine these concepts. Certainly the pre-Covid motivations for this text remain in place: the atmospheric concentration of 412 ppm (as of August 2020) is deplorably (but unsurprisingly) increased from that reported in Figure 2.1, as is the parallel change to ocean acidity (Section 12.1). This edition introduces three completely new chapters to the text: 1. Chapter 13 stems directly from the Covid-19 pandemic of 2020, a whole chapter focusing on pandemic modelling and on the challenges that complex systems present to policy-makers and healthcare. 2. Appendix C is a twenty-page extension to the spring-mass mechanical system of Case Study 5. The previous edition had repeatedly made claims about the role of eigendecompositions in decoupling systems, however, had offered only a single example of Case Study 5. Appendix C now gives six different examples, fol- lowed by a much more explicit discussion of precisely why it is that the eigendecomposition does not decouple nonlinear systems. 3. Appendix E offers a whole chapter providing solutions or summaries to a great many end of chapter problems, for students to be able to have feedback on their own attempts to solve problems. v vi Preface to the Second Edition (2020) A signiﬁcant part of my teaching pedagogy is to connect the dots in students’ minds, to reach a deeper and more nuanced understanding of complex systems. The eight new examples in this edition can be grouped into four contexts which mirror my own motivations in teaching: Examples 3.8 and 7.10 emphasize a breadth of understanding, connecting complex systems to large-scale questions, such as the systems-level questions in how to undertake a worldwide energy transition from fossil-based to fossil-free. Examples 4.4 and 10.3 develop a depth of understanding in some of the technical/mathematical aspects of complex systems, these two examples based on topics which students found confusing, speciﬁcally statistical stationarity (Example 4.4), and a better visualization of the critical phenomena in random graphs (Example 10.3). Given my own background in Electrical Engineering, I found it a bit frustrating how few circuit-related examples had ended up in the text. In particular, it is “widely known” that linear RLC1 circuits can be decoupled via eigendecom- positions, however, there are precious few examples of this idea to be found (really anywhere), so Example 7.4 (together with Problem 7.15 and Section C.4), emphasizes the decoupling of linear circuits, whereas Example 7.8, looks at a simple2 nonlinear oscillator. The one thing that makes studying complex systems fun is the ubiquity of the topic, so Examples 7.6 and 10.1 (and also Problems 6.10, 6.11), explore a more whimsical side, how the topics of this text routinely appear in the world around us (Example 7.6), sometimes in the most unexpected places, such as chocolate (Example 10.1). I would like to express my thanks to those students and colleagues who provided feedback on parts of this revised text. In particular, I would like to recognize the helpful comments from Nicholas Pellegrino and Tyron Jung. Finally, as always, my appreciation for the ongoing love, support, and enthu- siasm from my wife and children! Waterloo, Canada Paul Fieguth 1 Resistor-Inductor-Capacitor circuits. 2 The circuit is very simple to build, but not at all simple to simulate or analyze. I found it remarkable that my electronic set, which I had as a child, listed a variety of simple buzzer/oscillator circuits, easily built in a minute or two as a ten-year-old, yet in four years of studying electrical engineering I was not taught (or even had mention of!) a single oscillator circuit. Analyzing transistor oscillators really is hard—the circuit is nonlinear—but omitting it entirely leaves the student with a signiﬁcant gap in their understanding. Preface to the First Edition (2016) Although I had studied nonlinear dynamic systems, bifurcations, and bi-stable systems as a graduate student, I had clearly never really absorbed the ideas at a conceptual level, since it was ten years later that I was struck by a simple ﬁgure of catastrophic nonlinear state transitions ﬁg-fold in the context of ecological systems. I was immediately hooked: The concept was so clear, so elegant, and so easy to understand. How was it possible that I had never really encountered this? Over time, I became convinced that not just I, but indeed most of the under- graduate students with whom I interact, fundamentally do not grasp the big picture of the issues surrounding them, even though the underlying mathematical concepts are well within reach. The underlying problem is clear: for pedagogical reasons nearly all of the courses which my students take focus on mathematics and systems which are small scale, linear, and Gaussian. Unfortunately, there is not a single large-scale ecological or social phenomenon which is scalar, linear, and Gaussian. This is very simply, the rationale for this text: To explore a variety of large issues—global warming, ice ages, water, poverty—and to motivate and teach the material of the course—nonlinear systems, non-Gaussian statistics, spatial systems, complex systems—motivated by these case studies. The large-scale problems challenging the world are complex and multifaceted, and will not be solved by a single strategy, academic ﬁeld, or perspective. This book cannot claim to teach how to solve such enormous problems, however, the intent is very much to draw explicit parallels and connections between the math- ematical theory, on the one hand, and world issues/case studies on the other. The speciﬁc topics being taught are nonlinear dynamic systems, spatial systems, power-law systems, complex systems, and inverse problems. To be sure, these ﬁelds have been around for some time and many books have been written on the subjects, however, the ﬁelds are, at best, only weakly present in most undergraduate programs. This book is intended for readers having a technical background, such as engineering, computer science, mathematics, or environmental studies. The asso- ciated course which I have taught is open to third and fourth year undergraduate students, however, this book should, I hope, be of interest and mostly accessible to vii viii Preface to the First Edition (2016) a signiﬁcantly wider audience. The only actual prerequisites are some background in algebra and in probability & statistics, both of which are summarized in the appendices. The reader who would prefer to get a perspective of the text might prefer to ﬁrst read the two overview chapters, on Global Warming in Chapter 2, and on Water in Chapter 12. There are many online resources related to nonlinear dynamics and complex systems, however, online links can frequently change or become outdated, so I am reluctant to list such links here in the text. Instead, I am maintaining a web page associated with this book, at http://complex.uwaterloo.ca/text to which the reader is referred for further reading and other material. A number of people were of signiﬁcant support in the undertaking of this textbook. Most signiﬁcantly, I would like to thank my wife, Betty Pries, who was tireless in her enthusiasm and support for this project and regularly articulated the value which she perceived in it. My thanks to Prof. Andrea Scott and Dr. Werner Fieguth, both of whom read every page of the book from end to end and provided detailed feedback. Appreciation to Dr. Christoph Garbe, my host and research collaborator at the University of Heidelberg, where much of this text was written. Teaching this material to students at the University of Waterloo has allowed me to beneﬁt from their creative ideas. Here, I particularly need to recognize the contribution of the project reports of Maria Rodriguez Anton (discount function), Victor Gan (cities), Kirsten Robinson (resilience), Douglas Swanson (SOC control), and Patrick Tardif (Zipf’s law). Finally, my thanks to the contributions of my children: Anya, for allowing her artwork to appear in print in Figure A.2; Thomas, for posing at Versailles and appearing in Example 3.1; Stefan, for demonstrating an inverted pendulum in Figure 5.8. Waterloo, Ontario, Canada Paul Fieguth Contents 1 Introduction.......................................... 1 1.1 How to Read This Text.............................. 2 References............................................ 4 2 Global Warming and Climate Change...................... 5 Further Reading........................................ 13 References............................................ 13 3 Systems Theory........................................ 15 3.1 Systems and Boundaries............................. 16 3.2 Systems and Thermodynamics......................... 21 3.3 Systems of Systems................................ 25 Case Study 3: Nutrient Flows, Irrigation, and Desertiﬁcation........ 32 Further Reading........................................ 36 Sample Problems (Partial Solutions in Appendix E).............. 37 References............................................ 41 4 Dynamic Systems...................................... 43 4.1 System State...................................... 44 4.2 Randomness...................................... 48 4.3 Analysis......................................... 49 4.3.1 Correlation................................. 49 4.3.2 Stationarity................................ 55 4.3.3 Transformations............................. 57 Case Study 4: Water Levels of the Oceans and Great Lakes........ 60 Further Reading........................................ 62 Sample Problems (Partial Solutions in Appendix E).............. 63 References............................................ 66 ix x Contents 5 Linear Systems........................................ 67 5.1 Linearity........................................ 68 5.2 Modes.......................................... 70 5.3 System Coupling.................................. 72 5.4 Dynamics........................................ 75 5.5 Control of Dynamic Systems.......................... 81 5.6 Non-Normal Systems............................... 84 Case Study 5: System Decoupling........................... 86 Further Reading........................................ 91 Sample Problems (Partial Solutions in Appendix E).............. 92 References............................................ 98 6 Nonlinear Dynamic Systems—Uncoupled.................... 99 6.1 Simple Dynamics.................................. 100 6.2 Bifurcations...................................... 105 6.3 Hysteresis and Catastrophes........................... 111 6.4 System Behaviour Near Folds......................... 120 6.5 Overview........................................ 122 Case Study 6: Climate and Hysteresis........................ 124 Further Reading........................................ 128 Sample Problems (Partial Solutions in Appendix E).............. 132 References............................................ 141 7 Nonlinear Dynamic Systems—Coupled...................... 143 7.1 Linearization...................................... 144 7.2 2D Nonlinear Systems............................... 147 7.3 Limit Cycles and Bifurcations......................... 151 7.4 Control and Stabilization............................. 162 Case Study 7: Geysers, Earthquakes, and Limit Cycles............ 164 Further Reading........................................ 174 Sample Problems (Partial Solutions in Appendix E).............. 175 References............................................ 185 8 Spatial Systems........................................ 187 8.1 PDEs........................................... 189 8.2 PDEs & Earth Systems.............................. 192 8.3 Discretization..................................... 195 8.4 Spatial Continuous-State Models....................... 200 8.5 Spatial Discrete-State Models......................... 208 8.6 Agent Models..................................... 214 Case Study 8: Global Circulation Models...................... 218 Further Reading........................................ 221 Sample Problems (Partial Solutions in Appendix E).............. 222 References............................................ 231 Contents xi 9 Power Laws and Non-Gaussian Systems..................... 233 9.1 The Gaussian Distribution............................ 234 9.2 The Exponential Distribution.......................... 235 9.3 Heavy Tailed Distributions........................... 238 9.4 Sources of Power Laws.............................. 247 9.5 Synthesis and Analysis of Power Laws.................. 250 Case Study 9: Power Laws in Social Systems.................. 257 Further Reading........................................ 261 Sample Problems (Partial Solutions in Appendix E).............. 261 References............................................ 267 10 Complex Systems...................................... 269 10.1 Spatial Nonlinear Models............................ 270 10.2 Self-Organized Criticality............................ 278 10.3 Emergence....................................... 285 10.4 Systems of Complex Systems......................... 288 Case Study 10: Complex Systems in Nature.................... 289 Further Reading........................................ 291 Sample Problems (Partial Solutions in Appendix E).............. 292 References............................................ 298 11 Observation and Inference............................... 299 11.1 Forward Models................................... 300 11.2 Remote Measurement............................... 303 11.3 Resolution....................................... 307 11.4 Inverse Problems.................................. 316 Case Study 11A: Sensing—Synthetic Aperture Radar............. 329 Case Study 11B: Inversion—Atmospheric Temperature........... 333 Further Reading........................................ 334 Sample Problems (Partial Solutions in Appendix E).............. 335 References............................................ 341 12 Water............................................... 343 12.1 Ocean Acidiﬁcation................................. 345 12.2 Ocean Garbage.................................... 346 12.3 Groundwater...................................... 347 Case Study 12: Satellite Remote Sensing of the Ocean............ 350 Further Reading........................................ 352 Sample Problems....................................... 352 References............................................ 353 13 Pandemics and Complex Systems.......................... 355 13.1 Models of Virus Spread............................. 356 13.2 Changing Pandemic Dynamics......................... 360 13.3 Complex Systems and Healthcare Responses.............. 366 xii Contents Further Reading........................................ 371 Sample Problems (Partial Solutions in Appendix E).............. 371 14 Concluding Thoughts................................... 375 Further Reading........................................ 376 Sample Problems (Partial Solutions in Appendix E).............. 377 References............................................ 378 Appendix A: Matrix Algebra................................... 379 Appendix B: Random Variables and Statistics..................... 389 Appendix C: System Decoupling................................ 397 Appendix D: Notation Overview................................ 419 Appendix E: End of Chapter Solutions........................... 423 Index...................................................... 459 List of Examples Case Study 3 Nutrient Flows, Irrigation, and Desertiﬁcation......... 32 Case Study 4 Water Levels of the Oceans and Great Lakes......... 60 Case Study 5 System Decoupling............................. 86 Case Study 6 Climate and Hysteresis.......................... 124 Case Study 7 Geysers, Earthquakes, and Limit Cycles............. 164 Case Study 8 Global Circulation Models....................... 218 Case Study 9 Power Laws in Social Systems.................... 257 Case Study 10 Complex Systems in Nature...................... 289 Case Study 11A Sensing—Synthetic Aperture Radar................. 329 Case Study 11B Inversion—Atmospheric Temperature............... 333 Case Study 12 Satellite Remote Sensing of the Ocean.............. 350 Example 3.1 Three systems and their envelopes................. 17 Example 3.2 Complete Accounting........................... 20 Example 3.3 Entropy Reduction.............................. 23 Example 3.4 Society, Civilization, and Complexity............... 26 Example 3.5 Energy Returned on Energy Invested (EROEI)........ 27 Example 3.6 Maximum Power Principle....................... 29 Example 3.7 Global Flows I................................ 30 Example 3.8 Systems and Energy Transitions................... 31 Example 4.1 System Examples.............................. 47 Example 4.2 Correlation Lag—Mechanical System............... 53 Example 4.3 Correlation Lag—Thermal System................. 54 Example 4.4 Model and Residual Stationarity................... 58 Example 5.1 The Exponential Function........................ 73 Example 5.2 Coupled Systems............................... 76 Example 5.3 Resilience I—Linear Systems..................... 87 Example 6.1 Population Growth............................. 104 Example 6.2 Bead on a Hoop............................... 108 Example 6.3 System Perturbations............................ 115 Example 6.4 Deliberate Hystereses........................... 117 xiii xiv List of Examples Example 6.5 Playing with Blocks............................ 118 Example 6.6 Hysteresis in Freshwater Lakes.................... 119 Example 7.1 Nonlinear Competing Dynamics................... 152 Example 7.2 Basins for Root Finding......................... 153 Example 7.3 Resilience II—Nonlinear Systems.................. 154 Example 7.4 Dynamics of Linear Circuits...................... 155 Example 7.5 Creating and Finding a Limit Cycle................ 156 Example 7.6 Tapping, Banging, Ringing, and Whistling must all be Nonlinear................................ 157 Example 7.7 An Alarm Bell as Limit Cycle.................... 158 Example 7.8 Dynamics of Nonlinear Circuits................... 159 Example 7.9 Limit Cycles in Economics....................... 165 Example 7.10 Low Hanging Fruit and System Limits.............. 166 Example 7.11 The Tragedy of the Commons..................... 167 Example 7.12 Societal Dynamics.............................. 170 Example 8.1 Spatial Discretization............................ 205 Example 8.2 The Spatial Dynamics of Human Impact............. 206 Example 8.3 Two-Box Lumped-Parameter Model................ 209 Example 8.4 Global Flows II................................ 210 Example 8.5 Spatial Nonlinear Systems & Trafﬁc Jams............ 219 Example 9.1 Memory and Exponentials........................ 237 Example 9.2 Power Laws in Finance.......................... 245 Example 9.3 Power Laws in Weather......................... 248 Example 9.4 Power Laws and Fraud Statistics................... 251 Example 9.5 Power Laws and Cities.......................... 258 Example 9.6 Power Laws and Discount Functions................ 259 Example 10.1 Even Chocolate is Nonlinear!..................... 273 Example 10.2 Ferromagnetic Phase Change...................... 275 Example 10.3 Critical Behaviour in Connected Graphs............. 279 Example 10.4 Spatial SOC—Earthquakes....................... 284 Example 10.5 Slime Mould Emergence......................... 286 Example 10.6 Resilience III—Complex Systems.................. 287 Example 11.1 Indirect Inference............................... 302 Example 11.2 Can a Space Telescope Read a Newspaper?.......... 315 Example 11.3 Measurements and Constraints.................... 320 Chapter 1 Introduction For every complex problem there is an answer that is clear, simple, and wrong Paraphrased from H. L. Mencken The world is a complex place, and simple strategies based on simple assumptions are just not sufficient. For very good reasons of pedagogy, the vast majority of systems concepts to which undergraduate students are exposed are Linear: Since there exist elegant analytical and algorithmic solutions to allow linear problems to be easily solved; Gaussian: Since Gaussian statistics emerge from the Central Limit Theorem, are well behaved, and lead to linear estimation problems; Small: Since high-dimensional problems are impractical to illustrate or solve on the blackboard, and large matrices too complex to invert. Unfortunately, nearly all major environmental, ecological, and social problems facing humankind are non-linear, non-Gaussian, and large. To be sure, significant research has been undertaken in the fields of large-scale, nonlinear and non-Gaussian problems, so a great deal is in fact known, however the analysis of such systems is really very challenging, so textbooks discussing these concepts are mostly at the level of graduate texts and research monographs. However there is a huge difference between analyzing a nonlinear system or deriving its behaviour, as opposed to understanding the consequences of a system being nonlinear, which is much simpler. It is the latter consequences which are the scope and aim of this book. That is, although a detailed analysis of a complex system is, in most cases, too difficult to consider teaching, the consequences of such systems are quite easily understood: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 1 P. Fieguth, An Introduction to Complex Systems, https://doi.org/10.1007/978-3-030-63168-0_1 2 1 Introduction A nonlinear system is subject to irreversibility, such that given some change in the inputs to the system, undoing the change does not necessarily return the system to its start, whereas all linear systems are reversible. Furthermore nonlinear systems can be subject to discontinuous or catastrophic state changes, which is not possible in linear systems. Non- Gaussian/power- law systems may be characterized by extreme behaviours which would appear to be unpredictable or unprecedented based on historical data. In contrast, Gaussian statistics converge reliably and effectively assign a probability of zero to extreme events, giving a false sense of security in those circumstances where the underlying behaviour is, in fact, power-law. Large coupled nonlinear spatial systems, also known as complex systems, can give rise to highly surprising macroscopic behaviour that would scarcely be recognizable from the microscopic model, a phenomenon known as emergence. Managing such systems, particularly in response to some sort of failure, is very difficult. In contrast, the nature of linear systems is unaffected by scale. The sorts of systems that we are talking about are not arcane or abstract, rather there are well-known systems, such as geysers and toy blocks, or indeed everyday systems such as stock markets or weather. It is precisely because such systems are so common and everyday that engineers or other technical professionals are likely to encounter them and need to be informed of the possible consequences of interacting with and influencing such systems. The context of this text is sketched, necessarily oversimplified, at a high level in Figure 1.1. Essentially we have two interacting classes of systems, the human/ societal/economic and natural/ecological/environmental systems, both of which will exhibit one or more elements of nonlinearity and spatial interaction which lead to complex-systems and power-law behaviours. Since a model alone is of limited utility, we are interested in performing inference by combining models with measurements, particularly global-scale remotely-sensed measurements from satellites. 1.1 How to Read This Text This text is aimed at a target audience of undergraduate engineering students, but is intended to be more broadly of interest and accessible. Those readers unfamiliar with this text, or with complex systems in general, may wish to begin with the overview in Chapter 2, followed by a survey of the case studies which are presented at the end of every chapter, and which are listed on page XIII. An explicit goal of this text is not to focus attention on the mathematics behind complex systems, but to develop an understanding of the interaction between com- plex systems theory and the real world, how complex systems properties actually manifest themselves. For this reason there are, in addition to the end-of-chapter case studies, a large number of examples, listed on page XIII, and those readers more 1.1 How to Read This Text 3 Impact Observation Natural Human Model Inference Model World Nonlinear Nonlinear World Spatial Spatial Power-Law Policy Power-Law Complex System Complex System Resources Fig. 1.1 An Overview: Human/societal/economic systems (right) draw resources from and have an impact on natural/ecological/environmental systems (left). Both domains contain many examples of systems which exhibit one or more elements of nonlinearity and spatial interaction, leading to complex-systems and power-law behaviours. Observations of one or more systems are combined with models in an inference process, leading to deeper insights and (ideally) better policy. The red portions are the focus of this text, with the blue, green, and grey components illustrating the broader, motivating context. interested in a high level or qualitative understanding of the book may want to start by focusing on these. For those readers interested in the mathematics and technical details, the chapters of the book are designed to be read in sequence, from beginning to end, although the spatial and power law chapters can be read somewhat independently from the preceding material, Chapter 4 through Chapter 7, on dynamics and nonlinear systems. Complex systems can be studied and understood from a variety of angles and levels of technical depth, and the suggested problems at the end of every chapter are intended to reflect this variety, in that there are problems which are mathematical/analytical, computational/numeric, reading/essay, and policy related. The intent is that most of this text, and nearly all of the examples and case studies, can be understood and appreciated without following the details of the mathematics. The technical details do assume some familiarity with linear algebra and probability theory, and for those readers who need a bit of a reminder, an overview of both topics is presented in Appendices A and B. This book is, to be sure, only an introduction, and there is a great deal more to explore. Directions for further reading are proposed at the end of every chapter, and the bibliography is organized, topically, by chapter. 4 1 Introduction References 1. C. Martenson, The Crash Course: The Unsustainable Future of our Economy, Energy, and Environment (Wiley, New York, 2011) 2. M. Scheffer, Critical Transitions in Nature and Society (Princeton University Press, Princeton, 2009) 3. A. Weisman, The World Without Us (Picador, 2007) 4. R. Wright, A Short History of Progress (House of Anansi Press, 2004) Chapter 2 Global Warming and Climate Change There are few global chal- lenges as widespread, as com- plex, as significant to our future, and politically as con- troversial as that of global warming. The goal of this chapter is, in essence, to motivate this textbook; to convince you, the reader, that very nearly all of the topics in this text need to be understood in order to grasp the subtleties of a subject such as global warming. However the specific problem of global warming is not at all unique, in this regard. That is, after all, the premise of this book: that there is a wide variety of ecological and social challenges, all of which are highly interdisciplinary, and for which a person unfamiliar with one or more of systems theory, nonlinear dynamics, non-Gaussian statistics, and inverse problems is simply ill-equipped to understand. Other similarly broad problems would include Ecological pressures and extinction, Human Poverty, Energy, and Water, to which we shall return in Chapter 12. Allow me now to take you on a tour of the entire book, through the lens of global warming. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 5 P. Fieguth, An Introduction to Complex Systems, https://doi.org/10.1007/978-3-030-63168-0_2 6 2 Global Warming and Climate Change Chapter 3: Systems Theory Global warming is the warming of the earth’s climate caused by an increase in the concentrations of carbon-dioxide and methane gases in the atmosphere, due to human industry, fossil fuel consumption, and land use. Suppose we begin with a rather naïve model of carbon flows: ∞ Carbon Carbon ∞ Source Society Sink Energy Waste Human society, as a system, interacts with the rest of the world through the boundaries of the system. The naïve model has simplistically assumed fixed bound- ary conditions, that the sources (energy) and sinks (carbon pollution) are infinite, implying that humans do not influence the global climate. A slightly more realistic model, but also more complicated, understands energy sources and carbon-dioxide sinks to be finite, and therefore subject to influence by human activity: Finite Carbon Carbon Finite Society Source Energy Waste Sink At least two relatively significant problems remain present in this model: 1. The carbon does not just disappear into some abstract sink: the sink itself is subject to important dynamics. Carbon dioxide may be emitted into the atmo- sphere, however a great deal of it is absorbed into the ocean. Furthermore the ocean, itself, is far from being homogeneous and well-mixed, and the absorbed carbon will concentrate near the surface, whereas most of the potential for stor- age/sequestration is at the ocean bottom. 2. The size of the source or sink is not the only thing which matters: the flow rates are crucial to understand as well, in particular the flow rate relative to the size of a source or sink, or inconsistencies in flow rates from one system to another. The latter effect is very much present here, in that the flow rate of carbon from society into the atmosphere is controlled by humans, whereas the flow rate of carbon from the atmosphere is controlled by physical/natural processes. So although the ocean bottom provides an infinite carbon sink, in principle, inconsistencies in flow rates are causing carbon dioxide to continue to build up: 2 Global Warming and Climate Change 7 Fig. 2.1 Atmospheric Carbon Dioxide: CO2 400 Atmospheric CO (ppm) levels in the atmosphere have seen a steady and worrying increase over sixty years. 380 2 The annual variability is a seasonal effect, since there 360 are more forests in the northern hemisphere than in the southern. 340 320 1950 1960 1970 1980 1990 2000 2010 2020 Year Finite Carbon Carbon Medium Society Atmos. CO2 sink Source Energy Waste Rate Limited: Physics High Rate: Humans Ocean Large Surface CO2 sink Rate Limited: Nature Ocean Near ∞ Bottom CO2 sink So we cannot understand a system, such as a human society, in the absence of the interactions that the system has with others around it. Chapter 4: Dynamic Systems An additional subtlety is that the latter model is implicitly time-dynamic: there are rates of carbon transfer, and a continual transfer of carbon into a finite system implies a system which changes over time. Two time-dynamic changes associated with carbon are indisputable: 1. CO2 concentrations in the atmosphere are increasing over time, from 315ppm in 1960 to over 400ppm today (Figure 2.1); 2. Since 1800 the world’s upper oceans have increased in acidity by 0.1 pH point, corresponding to a 30% increase in acidity, due to the uptake of CO2 into the ocean and the formation of carbonic acid , discussed further in Section 12.1. 8 2 Global Warming and Climate Change On the other hand, when we try to articulate questions of global warming, or any other sort of climate change over time, the problem becomes much more slippery: What is our Baseline? We know that the climate has always been changing, switching between epochs with and without ice ages, for example. Indeed, the earth’s climate changes over time periods of all scales: Tens of millions of years … Tropical Jurassic (dinosaur) era Tens of thousands of years … Ice ages Hundreds of years … Medieval warming period (950–1250 AD), So-called little ice age (1550–1850 AD) Years … 1930’s dust bowl, El-Niña / El-Niño So at what point in time do we actually start measuring, in order to know whether the earth is warming or not? What is Actually Warming? The atmosphere is well-mixed and relatively easy to measure, whereas the heat uptake patterns in the oceans and the ground are far more variable, depending on the presence of subsident currents in the ocean, or geothermal activity underground. On the other hand, although land and ocean have a far greater mass than the atmosphere, it is primarily the upper surfaces of land and ocean which would be warming, only a small fraction of the total: Atmosphere 6 · 1014 kg per metre at sea level, 5 · 1018 kg in total Ocean 3 · 1017 kg per metre of depth, 1 · 1021 kg in total 17 Land 4 · 10 kg per metre of depth, Ill-defined total mass Furthermore it is primarily the upper surfaces of land and ocean which are biolog- ically active, and thus a warming of only this top sliver may produce a dispropor- tionate ecological impact. So is global warming … 1. a physical concept, a warming of the land-oceans-atmosphere in total, which can be measured as increases in global average temperature?, or 2. an environmental concept, a disturbance to ecological balance caused by the warming of some part of the land-oceans-atmosphere, where the localized warming may produce almost no impact on the global average temperature, and so must be assessed indirectly via some other measurement? What is Causing the Warming? The increase, over time, in human fossil fuel consumption is well documented. Similarly the increase, over a similar period of time, in atmospheric CO2 concentration has been accurately measured. However a correlation, a statistical relationship, between two events is not the same as cau- sation, where one event can be said to cause another. Measuring a correlation is an exceptionally simple statistical test, whereas causa- tion requires a much deeper understanding. In the case of global warming we need 2 Global Warming and Climate Change 9 Table 2.1 Spatial and Temporal Scales: Typical scales in space and time for structures in ice, water, and air System Typical Structure Spatial Scales Temporal Scales Ocean Surface Eddies 10 km–100 km Days–Months Ocean Mid-Depth Southern Oscillation 100 km–1000 km Years Ocean Deep Thermohaline 1000s km 1000 Years Atmos. Local Storms 1 km–100 km Hours Atmos. Nonlocal Pressure Systems 1000 km Weeks Ice Local Cracks cm–km Seconds–Years Ice Nonlocal Sheets 1km–100 km 1–100 Years to understand the carbon cycle: the industrial and natural global sources and sinks of carbon. Chapter 7: Coupled Nonlinear Dynamic Systems On top of this, in trying to model the presence or flow of heat and energy present in the ocean-atmosphere system, over half of all kinetic energy is not in atmospheric storms or in ocean currents, rather in ocean eddies. However an eddy, like other similar forms of oceanic and atmospheric turbulence, is one of the hallmarks of coupled nonlinear systems, a nonlinear relationship between multiple elements. No linear system, regardless how complex or large, will exhibit turbulent behaviour, so we cannot fully model the ocean-atmosphere system without studying nonlinear systems. Chapter 8: Spatial Systems The atmosphere and oceans are not just nonlinear, or coupled, but indeed very large spatially. The governing equation for both air and water flow is the Navier–Stokes nonlinear partial differential equation:  Water/Ocean Navier–Stokes Small spatial details    Incompressible Slow changes over time   Challenging   Misfit Air/Atmosphere Navier-Stokes Large spatial scales    Compressible Fast changes over time with interesting challenges due to the different time/space scales of water and air. Really the modelling challenge is much worse, since the range of temporal and spatial scales is actually quite tremendous, as shown in Table 2.1, particularly when the near-fractal1 behaviour of ocean ice is included. 1 Meaning that there is a self-similar behaviour on a wide range of length scales, such as cracks in ice on all scales from nanometre to kilometres. 10 2 Global Warming and Climate Change Executing such a model has been and continues to be a huge numerical challenge: dense spatial grids, with layers in depth (ocean) or height (atmosphere) or thickness (ice) and over time. Chapter 11: System Inversion Now what would we do with such a model? Ideally we would initialize the model to the earth’s current state, and then run it forward to get an idea of future climatic behaviour: Now Simulation Time Model Spin Up Model to Present Prediction (Up to 1000s of years) (Up to 10s of years) (100s of years?) Drive with long term average Drive with observations No observations Such a process of coupling a model with real data is known as data assimilation or as an inverse problem. There is the true earth state z, which is unknown, but which can be observed or measured via a forward model C(): % & True state z(t) Observations m(t) = C z(t) + noise In principle what we want to solve is the inverse problem, estimating the unknown state of the earth from the measurements: Inverse Problem: find ẑ, an estimate of z, by inverting C(): ẑ = C −1 (m) (2.1) However almost certainly we cannot possibly obtain enough observations, especially of the deep oceans, to actually allow the inverse problem to be solved, analogous to an underconstrained linear system of equations. Instead, we perform data assimilation, incorporating the observations m into a simulation of a climate model having state z̃(t): We want to push the simulated state towards truth: z̃(t) z(t) The idea is to iteratively nudge z̃ in some direction to reduce ' % &' 'm(t) − C z̃(t) ' (2.2) where #·# measures inconsistency. Thus we are trying to push or nudge the simulation to be consistent with real-world measurements, and therefore hopefully towards the true real-world state. Chapter 11: System Sensing 2 Global Warming and Climate Change 11 So measurement is key to successfully modelling and predicting climate. What are the things we can actually measure: Atmospheric Temperature: Weather stations on the earth’s surface Weather balloons Commercial aircraft Satellite radiometers Oceanic Temperature: Satellite infrared measurements of ocean surface temperature Ocean surface height measurements (thermal expansion) Ocean sound speed measurements Buoys, drifters, gliders directly taking measurements in the ocean Temperature Proxies (indirect effects indicative of temperature) Arctic ice extent and number of ice-free days Arctic permafrost extent Date of tree budding / leaf-out / insect appearance / bird migrations If we consider global remote sensing via satellite, really the only possible mea- surement is of electromagnetic signals. Therefore a satellite is limited to measuring signal strength (brightness) and signal time (range or distance). The key idea, however, is that there are a great many phenomena z which affect an electromagnetic signal via a forward model C(), meaning that from the measured electromagnetic signals we can infer all manners of things, from soil moisture to tree species types to urban sprawl to ocean salinity, temperature, and currents. Chapter 9: Non-Gaussian Systems Given adequate observations to allow a model to be simulated and run, what do we do with the results? Understanding climate models and validating the simulated results are huge topics which could certainly fill another textbook. But even something much simpler, such as a time series of historical temperature data, can lead to challenges. All students are familiar with the Gaussian distribution,2 a distribution that very accurately describes the number of heads you might expect in tossing a handful of 100 coins. Phenomena which follow a Gaussian distribution, such as human height, are convenient to work with because you can take the average and obtain a meaningful number. This seems obvious. However most climate phenomena, and also a great many social systems, do not follow a Gaussian distribution, and are instead characterized by what is called a power law. Examples of power laws include meteor impacts, earthquake sizes, and book popularity. Given power law data (say of meteor impacts over the last ten years), the average is not representative of the possible extremes (think dinosaur extinction 2 Or bell curve or normal distribution; all refer to the same thing. 12 2 Global Warming and Climate Change Fig. 2.2 Stable climate states as a function of atmospheric carbon dioxide: The earth may have no ice (top) or may have polar ice to some latitude (middle); the arrows indicate the discontinuous climatic jumps (“catastrophes”). Although obviously an oversimplification of global climate, the effects illustrated here are nevertheless real. A more complete version of this diagram can be seen in Figure 6.19. …). In fact, taking an average over longer and longer periods of time still fails to converge. That seems strange. We humans only barely learn from history at the best of times; learning from historical power law data is even worse, because it is difficult to know how much data we need to reach a conclusion. Chapter 5: Linear and Nonlinear Systems Lastly, it is important to understand the macro-behaviour of climate as a nonlinear system: Linear systems are subject to superposition and have no hysteresis. Nonlinear systems are subject to catastrophes and hysteresis. These concepts are most effectively explained in the context of a plot of stable climate states, shown in Figure 2.2. The principle of superposition says that if increasing CO2 by some amount leads to a reduction in ice, then twice the CO2 leads to twice the reduction. Superposition is highly intuitive, very simple, and usually wrong. We are presently on the lower curve in Figure 2.2, a planet with a mixture of ice and water. As CO2 is increased, the amount of ice indeed slowly decreases, until point “A”, at which point an infinitesimal increase in CO2 leads to the complete disappearance of all ice. We have here a bi-stable nonlinear system with a bifurcation at point “A”, leading to a discontinuous state transition known as a “catastrophe.” In a linear system, to undo the climate damage we would need to reduce the CO2 level back to below “A”. A nonlinear system, in contrast, has memory, what is called hysteresis. Reducing CO2 to just below “A” has no effect, as we are stuck on the ice-free stable state; to return to the mixed water-ice stable state we need to reduce CO2 much, much further, to “B”. Chapter 10: Complex Systems 2 Global Warming and Climate Change 13 Climate, ecology, human wealth and poverty, energy, water are all complex sys- tems: nonlinear, non-Gaussian, coupled, poorly-measured, spatial dynamic prob- lems. Viewing any of these as systems with isolated or fixed boundaries, subject to superposition and Gaussian statistics, is to misrepresent the problem to such a degree as to render useless any proposed engineering, social, economic, or political solution. This book cannot pretend to solve global warming or any of many other com- plex problems, but perhaps we can take one or two steps towards understanding the subtleties of complex systems, as a first step in identifying meaningful solutions. Further Reading The references may be found at the end of each chapter. Also note that the textbook further reading page maintains updated references and links. Wikipedia Links: Global Warming, Climate Change A challenge, particularly with global warming, is that there is an enormous number of books, most polarized to the extremes of either denial or despair. Books such as What We Know About Climate Change , Global Warming Reader , or the personable Walden Warming offer a broad spectrum on the subject. Good starting points for global warming science would be the respective chapters in earth systems books, such as [3, 4]. Regarding the role of nonlinear systems in climate, the reader is referred to Case Study 6.5, the book by Scheffer , or the recent paper by Steffen et al.. References 1. K. Emanual, What We Know About Climate Change (MIT Press, 2012) 2. A. Johnson, N. White, Ocean acidification: The other climate change issue. Am. Sci. 102(1) (2014) 3. L. Kump et al., The Earth System (Prentice Hall, 2010) 4. F. Mackenzie, Our Changing Planet (Prentice Hall, 2011) 5. B. McKibben, The Global Warming Reader (OR Books, 2011) 6. R. Primack, Walden Warming (University of Chicago Press, Chicago, 2014) 7. M. Scheffer, Critical Transitions in Nature and Society (Princeton University Press, Princeton, 2009) 8. W. Steffen et al., Trajectories of the earth system in the anthropocene. Proc. Natl. Acad. Sci.33(115), (2018) Chapter 3 Systems Theory There are many very concrete topics of study in engineering and computer science, such as Ohm’s law in circuit theory or Newton’s laws in basic dynamics, which allow the subject to be first examined from a small, narrow context. We do not have this luxury. A topic such as global warming, from Chapter 2, is an inherently large, interconnected problem, as are topics such as urban sprawl, poverty, habitat destruction, and energy. For example the latter issue of fossil fuels and global energy limitations is truly interdisciplinary, requiring the understanding, in my opinion, of many fields: Electrical Engineering Energy Conversion Geophysics Physics Oil / Gas Reserves Thermodynamics Chemical Engineering Agriculture Ethanol, Biodiesel Food and Energy Environment Recent History Natural Resources Oil, Foreign Policy Statistics Data Analysis Ancient History Fall of Civilizations Finance Debt, Fiat money Sociology Economics Society and Change Prices, Shortages Psychology Demographics Human Behaviour Populations, Pensions Political science Governments and Change This cannot be studied as a set of small, isolated problems, since the interconnec- tions are what make the problem. In order to understand interconnected systems, we have to understand something regarding systems theory. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 15 P. Fieguth, An Introduction to Complex Systems, https://doi.org/10.1007/978-3-030-63168-0_3 16 3 Systems Theory 3.1 Systems and Boundaries A system is most fundamentally characterized by the manner in which it interacts with its surroundings: These basic systems are shown graphically in Figure 3.1. A quick illustration should make the definitions clear: Open: Most human systems are open, since virtually all human societies and com- panies are based around trade and the exchange of goods. Closed: The earth as a whole is very nearly closed, since the inputs to the earth are dominated by solar energy (with a tiny amount of mass transfer from meteors), and the outputs from the earth are dominated by thermal energy (with a small amount of mass from the upper atmosphere and the occasional space probe). Isolated: There are few, if any, isolated natural systems. Even a black hole is not isolated; to the contrary, it is quite open, since there can be substantial mass transfer into it. A well insulated, sealed test-tube in the lab would be close to being an isolated system. Every system has some boundary or envelope through which it interacts with the rest of the world. A significant challenge, then, is defining this boundary: What part of the greater system is placed inside the boundary, and modelled explicitly, and what is outside, ignored or modelled implicitly via the boundary? Interestingly, there is a tradeoff, illustrated in Figure 3.2, between the complexity of the represented system and the complexity of the boundary. This makes sense: Fig. 3.1 Basic System Categorization: Systems are distinguished on the basis of whether energy and/or mass can cross their boundaries 3.1 Systems and Boundaries 17 Example 3.1: Three systems and their envelopes The earth is very nearly a closed system, primarily sunlight coming in and thermal energy going out, with only tiny changes in mass: Sun TH = 6000K Space TC = 3K Gases Meteors Space Probes A single human is very much an open system, since food and water come into our bodies from outside of us. The goal in such a system diagram is to annotate the predominant interactions between a system and the outside world, not necessarily to be exhaustive and complete. Here, for example, we consider only movement of mass to/from a body, and not other interactions such as senses of touch, smell, sight, and hearing: Oxygen CO2 Water Vapour Liquid Water Heat Food Waste Human society is very open; indeed, the exchange of goods and trade is the hallmark of advanced societies. This diagram is terribly incomplete, as the range of materials coming into and out of a city is vast: Air Goods Pollution Pollution Energy Heat Water 18 3 Systems Theory Fig. 3.2 System versus Boundary Complexity: A self-contained system, right, may be relatively complex but has a simple boundary. Representing only a small part of a system, left, leaves us with a simple dynamic model, but the strong dependencies (thick lines) on unmodelled aspects of the system may leave us with a more difficult boundary representation than in the middle, where the dependencies are more modest (thin lines) Fig. 3.3 Boundary Conditions: An isolated system has no boundary dependencies. If the sources and sinks are infinite, then their behaviour is unaffected by the modelled system and the boundary is static. The most difficult circumstances are boundary interactions, right, with finite systems. The more of the system I move outside of the boundary, the simpler the remaining system, but the more complex the interactions through the boundary with the other parts of the system. The more of the system I move inside of the boundary, the more complex a system I have to model, but the simpler the interactions through the boundary. This is perhaps simpler to illustrate with an example: The Human Body is an enormously complex system, almost impossible to under- stand given the many organs and subsystems (circulatory, nervous etc.), however the envelope around the body is relatively simple: food, water, waste, air etc. A Cell is relatively simple, in contrast to the whole body, and a great many cell functions are understood in detail. However a cell has, at its boundary, a tremen- dously complex envelope of proteins, hormones, other molecules and physical forces induced by the body within which it lives. 3.1 Systems and Boundaries 19 Fig. 3.4 The heat flow Q across a temperature gradient from hot TH to cold TC allows a system S to do work W. The output heat flow Q C is commonly seen as waste energy, however attempting to do work on the basis of this waste, right, leads to a temperature drop across introduced system E, leading to a reduced gradient across S and a correspondingly reduced output W1 < W. Although there are clearly many possible ways of representing and modelling boundary conditions, very roughly there are three groupings, illustrated in Figure 3.3: No Boundary: Isolated systems, with no interactions modelled outside of the boundary. Static Boundary: Systems having infinite sources and sinks, meaning that the boundary conditions can simply be held fixed and are not influenced by the system in any way. Dynamic Boundary: Systems having finite sources and sinks, external systems which can be affected by the modelled system. In most circumstances the boundary between societal and environmental systems should fall into the third category, interacting with and affecting each other. However, as discussed in Example 3.2, given a societal system we often represent external environmental systems as being in one of the first two categories: either dismissed entirely (a so-called externality), or assumed to be fixed and immune from human influence, such as assuming the ocean to be an infinite carbon sink. The interaction between a system and its boundaries affects whether the system is able to perpetuate its activity. That is, a system is sustainable (in steady-state) if, for a given unchanging definition of system boundary, the system is able to continue its activity indefinitely. 20 3 Systems Theory Example 3.2: Complete Accounting Where do we draw system envelopes when it comes to economics and GDP calculations? In most current GDP calculations, the natural world is outside our system envelope: Wood Gold, Wood Natural Rest of World Society Human World Gold $$$ System GDP = $$$ The society has traded the mined gold for money. But after selling the gold, it is no longer in the ground! The gold in the ground was worth something, but that resource loss, or the fact that the forest takes time to regrow, is not accounted for. The gold and the forest are examples of externalities, aspects or components of a problem left outside of the system envelope. We may choose to limit the system envelope this way for a variety of plausible reasons: to limit model complexity, because of significant uncertainties in the external component, or because of standardized accounting practices. However just as often it may be for reasons of political expediency or accountability — if we don’t model or measure something (humanitarian issues, overfishing, clearcutting), then we can claim innocence or ignorance of the problem: out of sight, out of mind. Environmental or full-cost accounting extends the system boundaries to include the nat- ural world, to specifically account for removal losses and liabilities such as strip mining, clearcutting, and overfishing: Wood Gold, Wood Natural Rest of World Society Human World Gold $$$ System GDP = $$$ - Loss of Trees + Regrowth of Previously Cut Forest - Loss of Gold Ore - Pollution Due to Mining There are, of course, significant challenges and uncertainties associated with assigning “costs” to buried gold, water pollution, and extinct species. However such challenges should not stand in the way of including the natural world within our system envelope. Further Reading: There are many books on environmental accounting, such as , or. Also see Full-cost accounting or Environmental economics in Wikipedia. 3.2 Systems and Thermodynamics 21 3.2 Systems and Thermodynamics The ability of a system to do something, to be dynamic, relies on some sort of input–output gradient or difference: temperature, pressure, height, chemical etc. A turbine relies on a high–low pressure gradient, the human body relies on a food– waste chemical gradient, even a photovoltaic solar panel relies on an thermal-like gradient between the effective temperature of incoming sunlight (6000K) and that of the local environment (300K). The basic thermodynamic system in Figure 3.4 shows input energy Q H , waste heat Q C , and work W. By conservation of energy, we know that Q H = W + QC. (3.1) What interests us is to maximize W for a given amount of input energy Q H. Ther- modynamics places an upper limit on the efficiency of any system: ! " TC W ≤ Wopt = Q H 1 − (3.2) TH however this is an upper limit; it is certainly possible for W to be significantly below Wopt. We observe from (3.2) that a steeper gradient TH /TC leads to a more efficient system. It is for this reason that one needs to be careful in proposing schemes to harness “waste” energy: any additional system introduced on the “Cold” side requires a temperature gradient of its own, raising T̄C and reducing the efficiency of the original system, as shown in the right half of Figure 3.4. That being said, there are certainly many examples of successful waste heat recovery and co-generation where the waste heat is used directly, to heat a building, rather than to run a machine. Because of the limit to efficiency, it follows that Total input energy Q H #= Total usable energy W leading us to a definition for entropy: Entropy: A measure of the energy not available for doing work. In general, entropy is also a measure of energy dispersal: greater dispersal cor- responds to lower gradients, corresponding to lower efficiency and less work. It is essential to understand this dispersal effect; for example, it is widely quoted that Solar Energy Arriving on Earth 1017 W, Total Human Energy Use 1013 W. It is claimed, therefore, that there is no energy shortage, only a shortage of political will to invest in solar energy. However the 1017 W are exceptionally dilute (effectively a very low TH ), spread over the entire planet (including the oceans). To be useful, this energy needs to be collected / concentrated (high TH ) in some form: 22 3 Systems Theory Solar-Thermal energy Mirrors Solar-Electric energy Conversion, transmission Solar-Agricultural energy Harvest, transport, processing All of these require infrastructure and the investment of material and energy resources, a topic further discussed in Example 3.5. A dilute energy source is simply not equivalent to a concentrated one, even though both represent the same amount of energy. Although energy is always conserved, clearly useful energy is not.1 So two cups of water, one hot and one cold, present a temperature gradient that could perform work, an ability which is lost if the cups are mixed: Same Energy TH TC Tmid # $% & # $% & W >0 W = 0 ∆S > 0 (3.3) Energy is conserved, but the ability to perform work is not, therefore there is a cor- responding increase in entropy S. An equivalent illustration can be offered with a pressure gradient: Same (3.4) Energy PH PL PMid PMid # $% & # $% & W >0 W = 0 ∆S > 0 It is clear that gradients can spontaneously disappear, by mixing, but cannot sponta- neously appear.2 From this observation follows the second law of thermodynamics, that entropy cannot decrease in an isolated system. However, as is made clear in Example 3.3, although entropy cannot decrease in an isolated system, it can most certainly decrease in an open (crystallization) and 1 The ambiguity in definitions of useful or available energy has led to a proliferation of concepts: energy, free-energy, embodied-energy, entropy, emergy, exergy etc. I find the discussion clearer when limited to energy and entropy, however the reader should be aware that these other terms are widely used in the systems/energy literature. 2 There are phenomena such as Rogue Waves in the ocean and Shock Waves from supersonic flow, in which gradients do appear, however these are driven effects, induced by an energy input. 3.2 Systems and Thermodynamics 23 Example 3.3: Entropy Reduction The second law of thermodynamics tells us that entropy cannot decrease. Consider, then, the process of crystallization: Time Dissolved Solute Crystal (Disorder) (Order) We appear to have a spontaneous reduction in entropy, from disordered (high entropy) to ordered (low entropy)! However, considering a system envelope drawn around the beaker of solute, this is not an isolated system. Water has evaporated (S ) and crossed through the envelope, taking a great deal of entropy with it. So, indeed, the entropy inside the system envelope has decreased, however global entropy has increased, as expected. System boundaries matter! Fig. 3.5 In an isolated system entropy S cannot decrease, however entropy can most certainly decrease in open or closed systems, provided that sufficient entropy is pumped across the system boundary, the principle under which all refrigerators, freezers, and air conditioners work. 24 3 Systems Theory closed (fridge) systems, since open and closed systems allow energy (and entropy!) to cross their boundaries. As shown in Figure 3.5, non-isolated systems can import low-entropy inputs … high gradients (very hot and very cold), fuel, mechanical work, electricity and subsequently export high-entropy outputs … low gradients (warm thermal energy), burnt fuel, smoke, water vapour. So although a gradient cannot spontaneously appear, a refrigerator, for example, does indeed create a thermal gradient (cool fridge in a warm room). However to create this gradient requires it to consume an even greater gradient (energy) from somewhere else, and that energy (electricity) must have consumed an ever greater gradient (burning wood or coal) somewhere else, and so on. Furthermore the gradient metaphor generalizes to non-energetic contexts: to clean your hands (establishing a cleanliness gradient) requires you to consume an existing gradient, such as purified tap water or a clean towel; much more abstractly, entropy also applies to information, such as the computation or re-writing of a binary digit, known as Landauer’s principle. Very broadly, we have the following summary per- spective on entropy: Low Entropy High Entropy Life Death Order Disorder Construction Destruction Finally, not only do gradients not spontaneously appear, but nature exerts a force towards equilibrium and randomness: wood rots, metal rusts, stone weathers, animals die, things break. This tendency towards equilibrium consumes any kind of gradient, whether temperature (coffee cools, ice cream melts), pressure, chemical, cleanliness etc. However in order to do something, any form of work, a system cannot be in thermodynamic equilibrium. Life and all useful systems are therefore a striving towards maintaining gradients, holding off equilibrium, where ṠSystem (t) = ṠInput (t) − ṠOutput (t) + ṠNature (t) (3.5) So to maintain a given system complexity ( ṠSystem (t) ≤ 0) we need sufficiently low-entropy inputs (food, liquid water) and sufficiently high-entropy outputs (heat, water vapour, waste) to counter the natural push ( ṠNature (t) > 0) towards equilibrium (death). So for a system to be in steady-state, ṠSystem (t) = 0 (3.6) 3.2 Systems and Thermodynamics 25 meaning that sufficient entropy (and associated energy) flows must be present; that is, a system must live within the means of its energy budget to remain sustainable. A system can temporarily exceed its energy budget (running up a balance on its credit card, so to speak), but will eventually be forced to re-adjust, as discussed in Example 3.4. 3.3 Systems of Systems In principle a single system could contain many internal pieces, and therefore could be arbitrarily complex. However it is usually simpler to think of a “system” as being uniform or self- contained, with a natural boundary or envelope. In this case several separate, but interacting, systems would be referred to as a “system of systems,” as summarized in Table 3.1. The heterogeneity of Systems of Systems make them a significant challenge, so we need to maintain a certain level of humility in attempting to model or analyze such systems. Nevertheless Systems of Systems are all around us, Creek or Stream—plankton, fish, plants, crustaceans, birds … Meadow—perennials, annuals, worms, insects, microbes, … Human Body—organs, blood, immune system, bacteria, … Automobile—drive train, suspension, climate control, lighting, … Major City—buildings, transportation, businesses, schools, … so at the very least we want to open our eyes to their presence. In most cases, Systems of Systems are nonlinear, complex systems which will be further examined in Chapter 10, particularly in Section 10.4. Table 3.1 Systems of Systems: An interacting collection of systems is not just a larger system. Heterogeneous collections of systems possess unique attributes and pose challenges, yet are common in ecological and social contexts. 26 3 Systems Theory Example 3.4: Society, Civilization, and Complexity Formally, thermodynamics applies to large collections of particles (fluids and gases), to heat flow, and to issues of energy and energy conversion. To the extent that the internal combustion engine, electricity production, and agricultural fertilizers are critical compo- nents of modern civilization, thermodynamics is certainly highly relevant to world issues. We should not really be applying principles of thermodynamics informally to collections of people and treating them as “particles”; nevertheless thermodynamics can give insights into large-scale complex systems, which includes human society. Our study of thermodynamics concluded with two facts: 1. Nature exerts a force towards increased entropy or, equivalently, towards increased randomness, on every part of every system. 2. Opposing an increase in entropy, possible only in a non-isolated system, requires a flow of energy or energy-equivalent (e.g., oil, natural gas, minerals, or some other gradient flow). Therefore the greater the complexity of a system, The greater the force towards randomness, The greater the energy flow required to maintain the integrity of the system. This phenomenon will be clear to anyone who has done wiring or plumbing: It is easy to cut copper wire or pipe to length, but you cannot uncut a pipe, and at some point you are left with lengths which simply do not reach from A to B. The copper must then be recycled and melted (energy flow) to form a new material. More buildings, more computers, more information to maintain means more things that break, wear out, and require energy to fix or replace. If human civilizations invariably move towards greater complexity, And if energy flows are bounded or limited, Then all civilizations must eventually collapse. That is, as formulated by Greer and discussed in other texts [2, 17, 20], waste production (entropy) is proportional to capital, so a certain rate of production is required to maintain the productive capital. If production is insufficient to maintain the capital base, then some capital is forcibly turned to waste, leading to further reductions in capital, and further reductions in production. How far a society collapses depends on whether the resource base is intact (a so-called maintenance collapse), such that a supply of resources can be used at some stage to rebuild capital; or whether resources have been overexploited, preventing or greatly limiting the rebuilding of productive capital (a catabolic collapse). Example continues … 3.3 Systems of Systems 27 Example 3.4: Society, Civilization, and Complexity (continued) To be sure, a wide variety of societal dynamics has been proposed, from logistic (sus- tainable), cyclic expansion / collapse, an indefinite delay of collapse due to continual technological innovations and, in the extreme, the singularity hypothesis, an immortality / brain-on-a-chip future. In each model, societal complexity (black) is plotted relative to a complexity constraint (dashed red). Complexity Time Complexity Time (Limited energy flow) (Limit exceeded & eroded) Logistic Model Predator-Prey Model Complexity Complexity Singularity Time Time (Limits pushed via technology) (Unlimited in finite time) Indefinite-Growth Model Singularity Model Further Reading: There are many good books on this subject; see [2, 7, 17, 20]. Example 3.5: Energy Returned on Energy Invested (EROEI) The world runs on energy. As we saw in Example 3.4, the energy crossing the boundary into a complex system matters a great deal. World oil and gas production figures quote primary production of energy, in terms of the oil and gas coming out of the ground. However all of that energy is not actually available to society, since some of that energy got expended to get the oil and gas out of the ground in the first place: Net Energy = Primary Energy − Energy Expended to obtain Primary Energy What really matters to a society is the ratio Primary Energy EROEI = Energy return on investment = Expended Energy Example continues … 28 3 Systems Theory Example 3.5: Energy Returned on Energy Invested (EROEI) (continued) There are, of course, other possible ratios. Most significantly, historically, would have been the Food Returned on Food Invested (FROFI), where the 90% fraction of society neces- sarily involved in agriculture pretty much limited what fraction remained to do everything else — literature, philosophy, government, engineering etc. With less than two percent of the population in western countries now involved in agri- culture, the food ratio is no longer a limiting factor on maintaining and growing society, in contrast to energy. Raw energy is easily quantified, and widely reported, but the assessment of expended energy is much harder. The case of ethanol is particularly striking, since both oil and ethanol production are reported, however there is absolutely no way that Oil Production + Ethanol Production is actually available to society as energy, since there are very significant amounts of energy used in the agricultural production of corn and later distillation of ethanol. EROEI assessments suggest numbers on the order of Classic Texas gusher EROEI ≈ 100 Saudi Arabian oil 20 Ocean deepwater oil 3 Canadian tar sands 1 to 3 Wind 10 to 20 Solar Photovoltaics 6 to 10 Ethanol 0 x and y positively correlated: increasing x implies increasing y ρx y = 1 x and y exactly fall on a line with positive slope The correlation between two time series is a relatively simple concept, yet it is very easy to misinterpret. There are four common challenges or subtleties to keep in mind any time a correlation is measured: 4.3 Analysis 51 Fig. 4.4 Correlation: The correlation measures the linear relationship between two random variables x and y. Fig. 4.5 Spurious Correlations: The (i, j) element in each grid plots the correlation coefficient ρi j between time series i and j. In actual fact, all of the time series are uncorrelated; the apparent positive and negative correlations are coincident, and it takes a certain number of data points (right) to distinguish between real and spurious correlations. Number of samples: It takes data to establish a correlation: more data means greater confidence. With too few data points, even unrelated data sets can coinci- dentally appear to be correlated, an issue which is illustrated in Figure 4.5. There are formal statistical tests for this question (Pearson statistic, Chi-squared tests), and the reader is referred to the further reading (page 62) for references. At first glance the solution appears obvious, which is to acquire more data. However in practice the number of data points is very frequently limited: The data are collected over time, so more samples means more waiting; Data collection involves effort, so more samples means more money; Measuring instruments have a limited useful lifetime, and a given survey or field study runs only so long. For these and many other reasons it is actually quite common to encounter attempts to establish correlations in marginally short data sets. 52 4 Dynamic Systems Time lags: All real systems are causal, meaning that they cannot respond to an event until that event has taken place. So an abundance of food is only later reflected in an increased population, or a social welfare policy takes time to influence malnu- trition levels. Any sort of inertia, whether mechanical, thermal, or human behaviour, will lead to a delay in responding to an input. In assessing the correlation between input and response, we need to shift the response signal by some amount, such that (4.17) is changed to N , -, - 1 + x(i − δ) − µx y(i) − µ y ρx y (δ) = (4.18) N − 1 i=1 σx σy If, instead, we compute the lagged correlation of a signal with itself, N , -, - 1 + x(i − δ) − µx x(i) − µx ρx (δ) = (4.19) N − 1 i=1 σx σx we obtain what is known as the autocorrelation, which measures how quickly or slowly a signal decorrelates over time. Linear versus nonlinear: Correlation is only a measure of the linear relation- ship between two variables. However it is entirely plausible for two variables to be related, but nonlinearly so, such that there is no linear relationship, meaning that they are uncorrelated, even though the relationship could be exceptionally strong. One example of this effect is sketched in Figure 4.6. Detecting a nonlinear relationship is more challenging than calculating a correla- tion, since the form of the nonlinearity (sine, parabolic, exponential) needs to be known ahead of time. The human visual system is very capable at detecting such patterns, so in many case the best choice of action is the visual inspection of plotted data. Fig. 4.6 Correlation and Linearity: It is important to understand that correlation is a linear measure of dependence such that it is possible, right, to have random variables x and y which are strongly dependent but uncorrelated. 4.3 Analysis 53 Example 4.2: Correlation Lag — Mechanical System Let us consider the example of an oscillating mass m on a spring k, an example to which we will return in some detail in Case Study 5.6: k m Applied Force ξ(t) z(t) Suppose we had measurements of the applied force ξ, and we had an accelerometer and visual camera measuring the acceleration z̈, the velocity ż, and the location z of the mass. What might we expect to see? We know that.. ξ − kz z̈ = ż = z̈d t z = żd t (4.20) m Although there is an instantaneous relationship between force ξ and acceleration z̈, the inertia of the mass leads to a lag between force and velocity ż or position z, as made clear in the simulation: ξ(t) z̈(t) ż(t) z(t) 0 t Correlation versus causation: Ultimately we would like to understand cause and effect: Are humans causing global warming? Does free trade lead to increased employment? However whereas there are definitive statistical tests to establish the statistical significance of a correlation, meaning that two time series are connected, it is not possible to establish causation, as illustrated in Figure 4.7. Causation can only be asserted2 on the basis of a model, such as a model for global carbon- dioxide sources, sinks, and flows, or an economic agent-based model representing businesses, consumers, and price signals. 2There has been significant recent work on causality, discussed briefly under further reading on page 62. 54 4 Dynamic Systems Example 4.3: Correlation Lag — Thermal System Given a heat flow h(t) into an object having a specific heat K , the temperature is given by. 1 t T (t) = h(τ ) dτ + c (4.21) K so the thermal inertia of the object causes a lag between heat input and the resulting temperature. For example, although planet Earth is fed a nearly constant stream of input energy from the sun, a given location on the Earth has time-varying radiation input due to seasons and changing length of day: 400 1.0 Average Solar Radiation 20 Correlation Coefficient Average Temperature 10 300 0.9 0 200 0.8 −10 100 −20 0.7 0 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 0 20 40 60 80 Month Days of Temperature Lag The left plot clearly shows the overall correlation between incoming radiation and the average temperature, however the peak in the lagged correlation, right, is at 35 days. The thermal inertia can also be observed within 20 1000 single days. Day-time is clearly warmer than night-time, however the warmest part of the day is 15 750 Solar Radiation Temperature normally not right at noon (or at 1pm, with day- light savings time), when the sun is strongest. 10 500 Instead, due to the thermal inertia of the ground 5 250 and the air, the peak in temperature is slightly later in the afternoon, as is very clearly revealed in 0 0 plots correlating input–heat and lagged resulting– Day 1 Day 2 Day 3 Day 4 Day 5 temperature: 0.6 0.6 Correlation Coefficient

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