Working inside the control-theoretic framework for understanding thermodynamics, I develop a systematic way to characterize thermodynamic theories via their compatibility with various notions of coarse-graining, which can be thought of as parametrizing an agent’s degree of control of a system’s degrees of freedom, and explore the features of those theories. Phenomenological thermodynamics is reconstructed via the ‘equilibration’ coarse-graining where a system is coarse-grained to a canonical distribution; finer-grained forms of thermodynamics differ from phenomenological thermodynamics only in that some states of a system possess a free energy that can be extracted by reversibly transforming the system (as close as possible) to a canonical distribution. Exceeding the limits of phenomenological thermodynamics thus requires both finer-grained control of a system and finer-grained information about its state. I consider the status of the second law in this framework, and distinguish two versions: the principle that entropy does not decrease, and the Kelvin/Clausius statements about the impossibility of transforming heat to work, or moving heat from a cold body to a hotter body, in a cyclic process. The former should be understood as relative to a coarse-graining, and can be violated given finer control than that coarse-graining permits; the latter is absolute and binds any thermodynamic theory compatible with the laws of physics, even the entirely reversible limit where no coarse-graining is appealed to at all. I illustrate these points via a discussion of Maxwell’s demon.

This chapter deals with processes both from a macroscopic, thermodynamic point of view and from a dynamical perspective. For the latter, a class of processes is introduced that can be described through a Hamiltonian description with a time-dependent external control parameter. It is shown how the expressions of work and heat from classical thermodynamics can be obtained as an appropriate average over an initial distribution. The second law inequality relating work and free energy can then be proven as a consequence of a master inequality. With well-specified additional assumptions, second law inequalities for heat exchange and entropy production are derived.

Anti-evolutionists sometimes use principles of thermodynamics in making their case. Though thermodynamics is normally considered a branch of physics, it has a strongly mathematical character that justifies its inclusion in this book. We discuss the basics of thermodynamics and statistical mechanics, and then explain why the anti-evolutionist version is such a caricature.