This chapter follows a logic of exposition initiated by Gibbs in 1902. On the one hand, some theoretical results in statistical mechanics have been derived in Chapter 3, while, on another hand, some theoretical/experimental results are expressed within thermodynamics, and parallels are drawn between the two approaches. To this end, the theory of thermodynamics and its laws are presented. The chapter takes an approach where each stated law is attached to a readable source material and a person’s writing. The exposition of the second law follows the axiomatics of Carathéodory, for example. This has the advantage of decoupling the physics from the mathematics. The structure of thermodynamic theory with the scaling behaviour of thermodynamic variables, Massieu potentials and Legendre transformations is also developed. Finally, correspondence relations are postulated between thermodynamics and statistical mechanics, allowing one to interpret thermodynamic variables as observational states associated to certain probability laws. Applications are given, including the Gibbs paradox. The equivalence between the canonical and the microcanonical ensembles is analysed in detail.

This chapter traces the development of an economic sublime associated with modern neoclassical economics. A precursor to Fredric Jameson’s postmodern “hysterical sublime,” the sublimity of neoclassical economics derived from the thrilling sense that, through math, economics could access vast and terrifying universal forces, and connect individuals directly to them. However, the most popular expression of the economic sublime was not mathematical but literary, and consisted primarily of naturalist novels that chronicled human encounters with economic laws allegedly so regular, universal, and inexorable that they amounted to a new branch of physics. Through readings of novels by Theodore Dreiser, Jack London, and Frank Norris, this chapter shows how literature helped train readers to understand neoclassical economics not just as natural, but also as an indispensable source of romantic pleasure.

One of life’s most fundamental revelations is change. Presenting the fascinating view that pattern is the manifestation of change, this unique book explores the science, mathematics, and philosophy of change and the ways in which they have come to inform our understanding of the world. Through discussions on chance and determinism, symmetry and invariance, information and entropy, quantum theory and paradox, the authors trace the history of science and bridge the gaps between mathematical, physical, and philosophical perspectives. Change as a foundational concept is deeply rooted in ancient Chinese thought, and this perspective is integrated into the narrative throughout, providing philosophical counterpoints to customary Western thought. Ultimately, this is a book about ideas. Intended for a wide audience, not so much as a book of answers, but rather an introduction to new ways of viewing the world.

For a system in contact with a heat bath, it is shown how the distribution of any observable follows from a microcanonical description for the isolated system consisting of the system of interest and heat bath. The weak coupling approximation then leads to the standard expression for the canonical distribution. Free energy, canonical entropy, and pressure are introduced. For large systems, the equivalence of this canonical description with the microcanonical one is shown. For systems in contact with a particle reservoir, the grand-canonical distribution is derived. If the weak coupling approximation does not hold, the corrections due to strong coupling are determined. In particular, internal energy, free energy ,and entropy are identified such that the usual relations for these thermodynamic potentials hold true even in strong coupling.

The chapter begins with the basic thermodynamic concepts that form the basis of high-speed flow theory, including a basic physical understanding of the second law of thermodynamics. This results in the ability to use the isentropic flow relationships in analyzing the properties of a compressible flow field, which results in the ability to analyze flow in a stream tube, and understand how a converging–diverging nozzle works. The basic relations for determining the change in flow properties across shock waves and expansion fans are developed, which make it possible to analyze flow fields using shock and expansion calculation methods. The basic relations for viscous flow are developed, leading to the relations for calculating the local skin-friction coefficient for a compressible boundary layer. The reader will then be able to understand the cause and effect of shock–boundary layer and shock–shock interactions. Finally, concepts for how flight vehicles are tested in wind tunnels are developed, which explains why it is difficult to fully model full-scale flight characteristics.

Chapter 6 builds upon the foundation of divergences from Chapter 5, advancing into entropies and relative entropies with an axiomatic approach, and the inclusion of the additivity axiom. The chapter delves into the classical and quantum relative entropies, establishing their core properties and revealing the significance of the KL-divergence introduced in Chapter 5, notably characterized by asymptotic continuity. Quantum relative entropies are addressed as generalizations of classical ones, with a focus on the conditions necessary for these measures in the quantum framework. Several variants of relative entropies are discussed, including Renyi relative entropies and their extensions to quantum domain such as the Petz quantum Renyi divergence, minimal quantum Renyi divergence, and the maximal quantum Renyi divergence. This discourse underlines the relevance of continuity and its relation to faithfulness in relative entropies. The concept of entropy is portrayed as a measure with a broad spectrum of interpretations and applications across fields, from thermodynamics and information theory to cosmology and economics.

Information processing is a process of uncertainty resolution. Information-theoretic constructs such as surprisal and entropy reflect the fine-grained probabilistic knowledge which people have accumulated over time. The information-theoretic constructs explain the extent of processing difficulty that people encounter, for example when comprehending language. Processing difficulty and cognitive effort in turn are a direct reflection of predictability.

The extensive thermodynamic variables of a fluid are introduced as the internal energy, volume, and number of molecules. The entropy is defined and also shown to be extensive. Taking the total derivative of the internal energy produces the first law of thermodynamics and defines the intensive parameters of temperature, pressure, and chemical potential. Changing variables from extensive variables to intensive variables is accomplished with the Legendre transform and defines alternative energies such as the Helmholtz free energy, enthalpy, and Gibbs free energy. Thermodynamic equilibrium requires that each element of a system have the same temperature, pressure, and chemical potential. For equilibrium to be stable, the material properties of each element must satisfy certain derived constraints. First-order phase transition are treated for a single-species system. Multispecies systems are treated and a widely used expression for how the chemical potentials of each species depend on the concentration of the species is derived. Chemical reactions are treated as is osmosis. The thermodynamics of solid systems is addressed along with mineral solubility in liquid solutions.

This chapter covers digital information sources in some depth. It provides intuition on the information content of a digital source and introduces the notion of redundancy. As a simple but important example, discrete memoryless sources are described. The concept of entropy is defined as a measure of the information content of a digital information source. The properties of entropy are studied, and the source-coding theorem for a discrete memoryless source is given. In the second part of the chapter, practical data compression algorithms are studied. Specifically, Huffman coding, which is an optimal data-compression algorithm when the source statistics are known, and Lempel–Ziv (LZ) and Lempel–Ziv–Welch (LZW) coding schemes, which are universal compression algorithms (not requiring the source statistics), are detailed.

Conspiracy theories explain anomalous events as the outcome of secret plots by small groups of people with malevolent aims. Is every conspiracy unique, or do they all share a common thread? That is, might conspiracy explanations stem from a higher-order belief that binds together a wide variety of overtly independent phenomena under a common umbrella? We can call this belief the conspiracy frame. Network science allows us to examine this frame at two different levels: by examining the structural coherence of individual conspiracies and by examining the higher-level interconnectivity of the conspiracy beliefs as a whole.