*A Dynamical Systems Theory of Thermodynamics* is a formidable monograph presenting the author's framework for thinking about thermodynamics from a dynamical systems perspective. While the book attempts to be self-contained, much of the content would not be accessible without a reasonably advanced background both in rigorous dynamical systems theory and mathematical thermodynamics. The monograph is at the level of an advanced text in mathematical physics, with the core of the book proceeding in a definition-theorem-proof format. While it contains many expository interludes, these are not especially pedagogical or illuminating regarding the mathematics, but instead serve to discuss broader issues in the physical interpretation of both the author's work, and its connection to other frameworks in thermodynamics. The book is useful as a good reference for a broad overview of both the history and contemporary status of many aspects of thermodynamics and statistical physics, though the bulk of the book is devoted to the author's specific model framework, and its extensions and implications.

The book begins with a reasonably comprehensive bird's-eye-view of the history of thermodynamics in Chapter 1, stretching from the ancient Greek schools of natural philosophy and mathematics to the contemporary research literature. The author gets across how contentious the field is, and how difficult the subject has been to develop systematically, in comparison to areas which were more easy to formalize and where there is a scientific consensus about the basic terminology. A good overview is presented of philosophical and technical problems at the foundations of thermodynamics, and how most contemporary formulations of the theory are unable to address all of these. Because of the breadth and depth, this overview is not easy to digest without either substantial prior knowledge of the mathematical foundations of thermodynamics or consistent use of more pedagogical references.

The second Chapter covers topics in general dynamical systems theory, with an emphasis on control-theoretic formulations of stability, controllability, and observability. Some concepts presented are standard, such as flows defined in terms of semigroups, Lyapunov stability, and monotone dynamical systems. Other aspects seem to be more niche, such as definitions of semistability as well as material on nonnegative dynamical systems. These ideas may be further exposited in previous books that the author has written, but unfortunately, this Chapter does not provide a thorough motivation for most of the mathematical objects discussed.

The core of the book, Chapters 3-15, presents the author's model of a mesoscopic compartmental dynamical system embodying ideas of energy flow between compartments. Much of this is presented in a proof-based manner, with an emphasis on rigorously establishing existence, uniqueness, and continuity properties of solution trajectories, equilibria, and various large-scale functions (entropy, ectropy, etc). The core model is presented in Chapter 3, with relationships to classical thermodynamics and kinetic theory (4-5), chemical kinetics (6), finite-time thermodynamics (7), phase transitions (8-10), spatially continuous (11), stochastic (12), and finally relativistic thermodynamics (13-15). Finally Chapters 16-18 are a series of more philosophical and broad conclusions and perspectives about thermodynamics, its cultural and philosophical aspects, and relations to the rest of science and humanity.

The book seems to present a mathematically consistent framework or model of a thermodynamic system that satisfies some important criteria desired from such systems. However, much of the mathematical presentation is quite sophisticated, employing tools from infinite-dimensional dynamical systems, PDE, stochastic analysis, delay-differential equations, as well as a development of the differential geometry of semi-Riemannian manifolds and Minkowskian spacetime (with some discussion of Einstein's field equations and its solutions). It is unclear how much of the model developed here is already in the peer-reviewed literature, and few examples are given of applications or uses of the framework in specific scientific contexts. The author does repeatedly show a relation to the entropy defined and the thermodynamic arrow of time and suggests that this framework satisfactorily addresses this and other foundational issues in thermodynamics.

The book suffers a bit from the author's choice of language, and excessively technical terminology makes it more difficult to read than necessary. As an example, the author discusses thermodynamic systems as (anti)cyclo-dissipative systems, which is defined only in other quite specialized literature. Additionally, a host of complicated terms are used which I had to look up as I came across them (e.g.~avulses, climacterical, pretermission, trammeled, zoecentric, and many others). While this makes the author's work sound very deep, many of these terms have plainer and clearer alternatives that would have made reading it substantially easier. The presentation is not very humble with respect to the model developed, suggesting that it unifies thermodynamics with numerous other scientific disciplines for the first time, and discussing how many great thinkers have been unable to do this. This contentious tone makes some portions of the book more difficult to accept on their own merits.

The overall organization and presentation of ideas is a bit strange, especially in terms of the use of footnotes and the authors' systematic approach to definitions. For example, algebraic groups are defined in Chapter 2 after semigroup theory has been presented and employed, and Abelian Lie groups are defined by a footnote in the Epilogue (Chapter 17) in a short diversionary discussion on symmetry breaking in electroweak theory. A few pages later in the Epilogue, the author defines fractals and fractal dimensions in a footnote while discussing relationships between entropy production and healthy vs cancerous aging in human physiology, implying that fractal dimension is a useful measure of physiological health in some tissues. Quantum mechanics, spin, and several aspects of condensed matter physics are mentioned in the introduction and final chapters, but never systematically described and related to thermodynamics like relativistic mechanics was in Chapters 13-15. In the Afterward (Chapter 18), the author defines foliations and discusses the difficulty of defining integrability in a footnote related to Parmenides' concept of a timeless universe and Einstein's general relativistic view of spacetime as a static object. The book has a Conclusion, an Epilogue (with its own conclusion section), and an Afterward. Together these three Chapters discuss a slew of socio-economic phenomena, neuroscience, origin of life theories, grand unified theories in fundamental physics, ancient and contemporary metaphysics, as well as the authors (strongly worded) opinions on modern University research and teaching, and the future of the human race.

Overall I think the monograph does provide some thought-provoking insights, and with its 478 references gives a thorough overview of contemporary and historical thermodynamics, as well as its relations to numerous other scientific and humanistic disciplines. While interesting to consider, I would suggest some of the book's broad claims be taken provisionally only after a thorough comparison with other literature.

Dr. Andrew Krause is a Departmental Lecturer in Applied Mathematics at the University of Oxford. His research is primarily in mathematical biology and nonlinear dynamical systems. More information about him can be found at

https://www.andrewkrause.org/.