In an increasingly specialized world, it is a rare pleasure when a book can build compelling and useful connections among widely different disciplines. This volume offers such a treat, merging physics (the mathematics of quantum field theory), computation (stochastic Petri nets), graph theory, electrical circuits, linear algebra, biology (epidemiology and wildlife ecology), and chemistry (reaction networks). Its central contribution is developing a detailed parallel between quantum field theory and systems governed by stochastic Petri nets (a domain the authors call “stochastic mechanics”). The latter chapters make clear that the authors have done much of their thinking in the context of chemistry, but it opens up a wide range of mathematical tools that will be of value in the study of decentralized distributed computing systems, such as agent-based systems, where different computing entities can join or leave the system over time.
Chapter 1 introduces the stochastic Petri net (SPN) with illustrations from wildlife ecology. Briefly, an SPN is a bipartite directed graph in which places (which can be marked with integers) alternate with transitions. To facilitate an ongoing illustration involving rabbits and wolves, the authors call places “species” and interpret the marking of a place as the number of members of that species that are currently alive. A transition is eligible to fire when all of its input places have nonzero markings, and will actually fire with a transition-specific probability. When it does fire, it reduces the population of each input species by one for each edge from the species to the transition, and increases the population of each output species by one for each edge from the transition to the species. Such a structure is isomorphic to a rate equation, a differential equation (or set of such equations) that specifies how the expected value of the markings in each place (the expected population of each species) varies over time, exemplified by the standard varieties of epidemiological models: SI, SIR, and SIRS. Chapter 2 discusses in detail how to derive the rate equation of an SPN from the SPN, and illustrates it from biology and chemistry.
Though the transitions in an SPN are stochastic, the rate equation is deterministic, yielding only expected values of the various species. Chapter 3 develops an orthogonal view that describes the time-dependent probability of various population configurations that can exist. The equation that governs the evolution of this vector is called the master equation of the system. It has the same general form as Schrödinger’s equation in quantum physics, except that the vector that evolves is of probabilities, while in quantum physics it is the wave function of complex amplitudes. Chapter 4 develops the parallels with quantum mathematics, and chapter 5 exhibits analogs of the annihilation and creation operators from quantum theory that allow one to express an analog to the Hamiltonian of a quantum system. Chapter 6 applies this perspective to one biological system, and chapter 7 applies Feynman diagrams to another.
The next five chapters use the relation between the rate equation and the master equation to motivate the Anderson-Craciun-Kurtz theorem (chapters 8 and 9), which relates equilibrium solutions to the rate equation to corresponding solutions to the master equation, and Noether’s theorem (chapters 10 and 12), which relates symmetries to conserved quantities in both classical and quantum systems. Chapter 11 recapitulates the growing parallels between quantum theory and stochastic mechanics.
While chemical reaction networks motivate a number of earlier examples, in chapters 13 through 20 they are the central focus of attention, motivating introduction of the Desargues graph (chapter 13), graph Laplacians (chapter 14), and the parallel between Dirichlet operators, that is, Hamiltonians that satisfy the requirements both of quantum systems (self-adjoint) and stochastic mechanics (infinitesimal stochastic), and electrical circuits of resistors (chapter 15). This discussion leads to the Perron-Frobenius theory (chapter 16) and the deficiency zero theorem of chemistry (chapters 17, 18, 20), as well as further applications of the Anderson-Craciun-Kurtz and Noether’s theorems (chapter 19). Chapters 21 through 23 lead the reader through a proof of the deficiency zero theorem. Chapter 24 extends Noether’s theorem to Dirichlet operators, while chapter 25 addresses the formal computational properties of SPNs viewed as symmetric monoidal categories.
The book is highly accessible, written in an informal style that leads the reader through the proofs of the theorems presented. Each chapter includes several worked exercises, making the book ideal for self-study. The organization in the early chapters follows a logical progression, but later chapters appear to be a collection of diverse observations that grew out of the basic system of rate equations and master equations and their expression using math from quantum theory. The book does not consider generalized SPNs, which also allow inhibitory edges that disable a transition by joining a nonzero place to the transition. The presence of such edges makes an SPN computationally equivalent to a Turing machine. It would be very interesting if the authors would extend their treatment to include this case (and perhaps even more interesting if there is some disciplined reason why such an extension is impossible).