This chapter presents an overview of the goals of universal biology. It is noted that biological systems are generally hierarchical as molecules-cells-organisms, where the components of each level are quite diverse. How such diversity arises and is maintained is discussed. We then discuss the possibility of understanding such biological systems with diverse components, and explore the possibility of macroscopic theory to reveal and formulate universal properties in living states, noting that robustness, plasticity, and activity are essential to life. Recalling the spirit (not the formulation) of thermodynamics, we explore the possibility of formulating a theory for characterizing universal properties in life, emphasizing macroscopic robustness at each level of the hierarchy and the importance of macro-micro consistency.

Our fundamental theories, that is, the quantum theory and general relativity, are invariant under time reversal. Only when we treat systems from the point of view of thermodynamics, that is, averaging between many subsystem components, an arrow of time emerges. The relation between thermodynamic and the quantum theory has been fertile, deeply explored and still a source of new investigations. The relation between the quantum theory and gravity, while it has not yet brought an established theory of quantum gravity, has certainly sparked in-depth analysis and tentative new theories. On the other hand, the connection between gravity and thermodynamics is less investigated and more puzzling. I review a selection of results in covariant thermodynamics, such as the construction of a covariant notion of thermal equilibrium by considering tripartite systems. I discuss how such construction requires a relational take on thermodynamics, similar to what happens in the quantum theory and in gravity.

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.

There is a long-cherished hope, which has its origins in the work of Boltzmann, that all that we are going to need to do in order to account for all the of the differences there are between the past and the future is to add to the fundamental time-reversal-symmetric dynamical laws, and to the standard statistical-mechanical probability-measure over the space of possible fundamental physical states, a simple postulate – a so-called “past hypothesis” – about the initial microstate of the universe as a whole. And there are various widespread and perennial sorts of puzzlement about how a hope like that can even seriously be entertained – puzzlements (that is) about how it is that the macrocondition of the universe 15 billion years ago, all by itself, can even imaginably be up to the job of explaining so much about the feel, today and on Earth, of the passing of time. I want to try to alleviate those puzzlements here. I will begin with a number of very general observations – and then, by way of illustration, I will present a new and detailed analysis of how it is that a simple pendulum clock invariably arranges to turn its hands clockwise in the temporal direction that points away from the Big Bang.

This chapter builds upon the previous chapters, applying the method of combining probability theory with Hamiltonian mechanics. To do so, one needs to build a meaningful sample space over states, in this case, quantum states. A substantial part of the chapter discusses how to construct these quantum states out of which one can build a sample space on which to apply a probability measure. Vector states and density operators are introduced and various worked examples are proposed. Once the quantum sample space is identified, the equilibrium quantum statistical mechanics is formulated. The ‘particle in a box’ problem turns out to be analytically intractable, unless we take a certain limit called the semi-classical limit. Heuristics as to what this limit means are proposed. Finally, the von Neumann (quantum) entropy is introduced and analogies with thermodynamics are made. An application to the heat capacity of solids is presented. As complement, the chapter also introduces a classical ‘ring-polymer’ analog of quantum statistical mechanics stating the formal equivalence between a one-particle quantum canonical system and an N-particle classical canonical system.

This short chapter aims at motivating the interest for statistical mechanics. It starts by a brief description of the historical context within which the theory has developed, and ponders its status, or lack thereof, in the public eye. A first original parallel of the use of statistics with mechanics is drawn in the context of error propagation analysis, which can also be treated within statistical mechanics. With regard to situations, statistical mechanics can be applied for, two categories are distinguished: experimental/protocol error or observational state underdetermining the mechanical state of the system. The rest of the chapter puts the emphasis on this latter category, and explains how statistical mechanics plays the role of ‘Rosetta Stone’ translating between different modes of description of the same system, thereby giving tools to infer relations between observational variables, for which we usually do not have any fundamental theory, from the physics of the underlying constituents, which is presumed to be that of Hamiltonian classical or quantum mechanics.

The chapter is an introduction to basic equilibrium aspects of phase transitions. It starts by reviewing thermodynamics and the thermodynamic description of phase transitions. Next, lattice models, such as the paradigmatic Ising model, are introduced as simple physical models that permit a mechano-statistical study of phase transitions from a more microscopic point of view. It is shown that the Ising model can quite faithfully describe many different systems after suitable interpretation of the lattice variables. Special emphasis is placed on the mean-field concept and the mean-field approximations. The deformable Ising model is then studied as an example that illustrates the interplay of different degrees of freedom. Subsequently, the Landau theory of phase transitions is introduced for continuous and first-order transitions, as well as critical and tricritical behaviour are analysed. Finally, scaling theories and the notion of universality within the framework of the renormalization group are briefly discussed.

Hopkins developed an ecological poetics informed by evolutionary theory, energy physics, and Catholic theology, bearing witness to local devastations of an unsustainable Victorian global economy. His sensitivity to such degradation was heightened by exposure to a range of polluted regions and by the effort to convey poetically his embodied perception of environmental features and patterns. His poems present everything from flashing bird wings, to waves, to wheat fields as dynamically interrelated through the flow of energy, and therefore vulnerable to its squandering by human industry. Such waste is both ecologically and spiritually self-destructive for Hopkins, given that Christ is incarnate in every fibre and force of the material world. His later sonnet ‘Ribblesdale’ manifests these concerns by lamenting a river valley poisoned and denuded by globally destructive industry and industrialized agriculture, even as it affirms vulnerable, accountable membership in a wounded terrestrial body that is divinely indwelt.

This chapter outlines Hopkins’s knowledge of contemporary energy physics as it decisively shapes his distinctive poetry and the metaphysic that undergirds it. The discussion begins with Hopkins’s appreciation of meteorology in his ‘Heraclitean Fire’ sonnet, of the earth’s atmosphere as a vast thermodynamic system. The figure that this poem presents of man as a lonely ‘spark,’ and the pyrotechnics of ‘As kingfishers catch fire,’ ‘The Windhover’ and ‘God’s Grandeur,’ are then glossed through the optical application of the energy concept in spectroscopy. Finally the chapter considers field theory and Clerk Maxwell’s reassessment of the Newtonian principle of force through the energy concept as the distributive principle of stress, tracing Hopkins’s use of this physical concept in his writings on mechanics, nature and most momentously in the definitive formulation of his metaphysic of stress, instress ,and inscape in 1868 and the concurrent advent of his metrical principle of Sprung Rhythm.

Chapter 1 introduces the first information measure – Shannon entropy. After studying its standard properties (chain rule, conditioning), we will briefly describe how one could arrive at its definition. We discuss axiomatic characterization, the historical development in statistical mechanics, as well as the underlying combinatorial foundation (“method of types”). We close the chapter with Han’s and Shearer’s inequalities, which both exploit the submodularity of entropy.