Review of Physics of Complex Systems: Discovery in the Age of Gödel by Dragutin Mihailović, Darko Kapor, Siniša Crvenković, and Anja Mihailović

In the preface, the authors state two guidelines that directed the composition of the book. The first guideline is the work of Gödel who expounded theories through reasoning opposite to accumulated experience and the mainstream. This is more for theoretical physicists and requires a deeper understanding of logic and mathematics. It seems that in the current context this will be beyond the reach of most hydrologists and will therefore be of limited value in hydrology and water engineering. The second guideline is the description and analysis of complex systems using information measures. There are several aspects presented in the book that are deemed relevant and useful in hydrology and some of the aspects are illustrated with hydrologic examples. Although the book is written for students in physics, this review highlights those concepts and theories that are or can be useful in hydrology.

The book is comprised of 10 chapters and two appendices. The first chapter is prolegomenon, which starts with a discussion of the generality of physics and goes on to discuss the crisis in physics, complex systems in physics, and association of physics and mathematics. The discussion in the chapter is a historical account, insightful and captivating, and is an interesting read even for a nonphysics major.

Chapter 2 is on Gödel’s incompleteness theorem and physics. Beginning with Gödel’s biography and a historical background of the incompleteness theorem, the chapter discusses an informal proof of the theorems, incompleteness theorems as a metaphor, and Gödel’s work in physical problems and computer science. The chapter is written lucidly but it is more for physicists, and hydrologists may not appreciate the theorems and the ensuing discussion.

Time in physics is the subject matter of Chapter 3. It begins with describing time in philosophy and physics, and goes on to discussing Gödel’s concept of time, thermodynamic arrow of time, quantum of time, and Planck time and Planck length. Thermodynamic arrow of time and Planck length and time may be of particular significance in hydrology. Thermodynamic arrow of time introduces the direction of time and entropy, which has found widespread application in water engineering. The chapter also discusses Planck length, which is formulated as the square root of the ratio of the product of Planck constant and gravitational constant to the cubic of speed of light and is equal to $1.616\times {10}^{-35}\mathrm{m}$. It seems there is a scope for defining a similar concept of length in hydrology—surface, subsurface, and groundwater—as well as water quality and hydraulics. In surface water hydrology, the concept of hydrologic response unit (HRU) is employed, and in numerical solutions, a grid in space and time is defined based on the Courant-Friedrichs-Lewy condition. It may be possible to use celerity in hydraulics and surface water hydrology, rate of mobility in subsurface hydrology, and pore velocity in groundwater in place of the speed of light. Of course, a constant similar to Planck’s constant will have to be sought. Looking at its dimensions of $\mathrm{L}/{\mathrm{T}}^2$, it seems it may be related to transmissivity in groundwater and diffusivity in subsurface and surface water hydrology as well as hydraulics. The Planck time is defined by the ratio of Planck length to speed of light. It may have an analogy with the time of travel in porous media. The chapter further discusses discrete and continuous time, time step, scaling of time, time in complex systems, and time in functional systems. Although the discussion in the chapter may seem esoteric, it does provide a wealth of information that may be insightful in hydrology.

Chapter 4 discusses the relation between theory and model in physics. Beginning with a discussion of background concepts and epistemology, it defines theory as a conceptualized framework of a phenomenon, and model as a physical representation of a concept to make the concept more understandable. It goes on to deal with choices in model building from microscopic to macroscopic: space-time in model building, Lyapunov time and Kolmogorov time in the predictability of complex systems, and chaos in environmental interfaces in climate change. This chapter is insightful and has a lot of value in hydrology. For example, complex systems are defined by many degrees of freedom and nonlinear interactions, and we model complex systems, not complexity. The relationship between a complex system and its host environment is regarded as critical to both model construction and model choice. The internal structure of the system can change dynamically owing to changes in the surrounding environment. This is directly related to nonstationarity, which is receiving a lot of attention in hydrology these days. Time and space play a key role in model building. It is stated that the discrete is not an approximation of the continuous, nor the converse. The prediction horizon is defined by the Lyapunov time, which is the inverse of the largest Lyapunov exponent. Deterministic chaos is another concept that is of direct application in hydrology. These concepts are of value in water engineering, especially in stochastic hydrology.

Inaccessible information is dealt with in Chapter 5. The physicality, abstractness, and information are discussed first. Physicality means that information has physical existence. Abstractness supposes that information is an idea that exists in our mind. Information can be either physical or abstract. Metaphysics of chance or probability is discussed next, including different interpretations of probability, such as classical probability, logical evidential probability, subjective probability, frequency interpretations, propensity interpretations, and best system interpretations, and explains the difference between chance and probability. The chapter then discusses Shannon information (the triangle of relationship between energy, mass, and information); the information that can be derived from rare events; and information in complex systems. This chapter is written well and is highly useful in hydrology.

Chapter 6 discusses Kolmogorov complexity and change complexities and their application in complex physical systems. As an incomputable measure but subject to approximation, the Kolmogorov complexity of an object represented by a finite binary sequence is defined by the minimum length of the binary sequences needed to define the whole object. Thus, an object is more complex if it is defined by a longer sequence and vice versa. Change complexity is a measure to detect change and can be applied to sequences of any length and its application in search for patterns in river flow, which is influenced by anthropogenic changes and climate change, and exhibits different levels of complexity, simple to complex. It demonstrates change complexity analysis for 10 streamflow time series. The chapter is written very well and contains several concepts that have a lot of potential for application in water engineering.

Separation of scales in complex systems is discussed in Chapter 7. It begins with the generalization of scaling, emergence and transitions between phases, scale invariance and universality, renormalization group, Heisenberg model, and complexity and time scale, the breaking point. When a phenomenon is modeled across a wide space-time range, a single model cannot capture it. Therefore, the range is separated into several scales. The chapter is full of information and most of the concepts discussed therein are useful in hydrology.

Chapter 8 discusses randomness and complexity of turbulent flows. The concept of energy cascade is at the core of the physics of turbulence and its multiscale nature. Kolmogorov defined turbulence as a self-similar cascade with universal properties. The Kolmogorov theory of turbulence is comprised of three hypotheses: local isotropy, first similarity hypothesis, and second similarity hypothesis. The concept of randomness is introduced as a result of uncertainty in the presence of determinism. Randomness and turbulent flow with Kolmogorov complexity, complexity of coherent structure in turbulent mixing layer, and information measures for describing river flow as a complex natural fluid system are presented. These concepts are fundamental, and their application can help unfold physical mechanisms in hydrology and hydraulics.

Physics of complex systems and art constitute the subject matter of Chapter 9. It discusses the complexity of the human brain, dualism between science and art, change complexity in psychology, entropy and change complexity, and Kolmogorov complexity in observing differences painting. Although applied to different fields, these concepts are useful in hydrology.

Modeling of biophysical systems is discussed in Chapter 10. It deals with the role of physics in modeling complex systems of the human body, stability in synchronization of intercellular communication in tissues, instability in synchronization of intercellular communication, and the search for information in brain disorders. The chapter is written well but is mainly for physicists.

The book is concluded with Appendix A on the mathematical derivation of the Lempel and Ziv algorithm and Kolmogorov complexity spectrum and Appendix B on the derivation of change complexity.

This is an excellent book containing a treasure trove of information. It reflects the rich and vast experience of authors. Even though the book is meant for physicists, it has plenty of material that is useful in hydrology and has the potential for application in water engineering. The authors deserve applause for writing the book.