This chapter discusses the notions of state of matter and phase of matter. It looks at two categories of ‘anomalous behaviours’ in thermodynamics: pressure plateaus in the isotherms of real gases, and the appearance of a magnetic state in ferromagnet. The former situation lends itself to a thermodynamic analysis with the van der Waals equation of state. A full analysis is proposed and the interpretation of the pressure plateau as stemming from the coexistence between two different phases at different densities is identified. Various laws, such as the latent heat law and Clapeyron’s law, are derived as well from thermodynamic theory. In the case of magnetism, there is no equation of state that would play a role analogous to the van der Waals equation of state. Statistical mechanics is required to understand the physics at play in the system. This is done by looking at the paradigmatic Ising model. The mean-field approach to this model is proposed and the existence of a ferromagnetic phase, breaking the underlying symmetry of the system, is observed.

This chapter lays the foundation of probability theory, which has a central role in statistical mechanics. It starts the exposition with Kolmogorov’s axioms of probability theory and develops the vocabulary through example cases. Some time is spent on sigma algebras and the role they play in probability theory, and more specifically to properly define random variables on the reals. In particular, the popular notion that ‘the probability for a real variable to take on a single value’ is critically analysed and contextualised. Indeed, there are situations in statistical mechanics where some mechanical variables on the reals do get a non-zero probability to take on a single value. Moments and cumulants are introduced, as well as the method of generating functions, which prepare the ground as efficacious tools for statistical mechanics. Finally, Jaynes’s least-biased distribution principle is introduced in order to obtain a priori probabilities given some constraints imposed on the system.

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.

This chapter is concerned with Gibbs’ statistical mechanics. It relies on developing the constraints imposed by Hamiltonian mechanics on the time evolution of a general probability density function in phase space. This is effectively done by using the notion of Hamiltonian flow and material derivative. Combining conservation of probability with Liouville’s theorem of Hamiltonian mechanics gives rise to Liouville’s equation, which is a cornerstone equation of both time-dependent and equilibrium statistical mechanics. From there on, the chapter focuses on equilibrium statistical mechanics and introduces the canonical and microcanonical Gibbs’ ensembles. The chapter takes a step-by-step approach where the main ideas are presented first for one particle in one dimension of space, and then reformulated in more increasingly more complex situations. Important properties such as the partition function acting as a moment generating function are derived and put in practice. Finally, a whole section is dedicated to little know works from Gibbs on statistical mechanics for identical particles. Finally, the grand canonical ensemble is also introduced.

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.