The chapter is fully dedicated to the theory of large deviations. To carry out the proof of the theorem and the actual computation of various distributions of large deviations, a detailed appendix is dedicated to the saddle point theorem to compute certain fundamental integrals, recurring in the theory. Lagrange transforms stem naturally from the large deviations theory, and we discuss their properties “in-line” for non-experts.

This is a rich chapter in which we delve into the study of the (weak and strong) laws of large numbers, and of the central limit theorem. The latter is first considered for sums of independent stochastic variables whose distributions have a finite variance, and then for variables with diverging variance. Several appendices report on both basic mathematical tools and lengthy details of computation. Among the first, the rules of variable change in probability are presented, Fourier and Laplace transforms are introduced, and their role as generating functionals of moments and cumulants, and the different kinds of convergence of stochastic functions are considered and exemplified.

Analysis of experimental data with several degrees of freedom is reported, starting from the Gaussian case, from the ground of the least-squares method, whose theory is detailed at the end of the chapter, for both independent and correlated data. The multi-dimensional versions of the reweighting method for unknown distributed data and of the bootstrap and the jackknife resampling methods are presented. How the possible correlation of multivariate data affects the methods is discussed and dealt with.

This chapter follows on from the previous chapter in quantum statistical mechanics but specialising on systems with identical particles. Using Gibbs prescritpion on generic states from Chapter 3, the occupation number representation is introduced. Constraints imposed on statistics by irreducible representations of the permutation group are discussed. These group-theoretic considerations are used to justify the use of Gentile’s parastatistics. Fermions and bosons are introduced as special cases of Gentile’s statistics, corresponding to the trivial representation of the permutation group for bosons, and the sign representation of the permutation group for fermions. Basic applications to fermions and bosons is given, including the Fermi–Dirac and Bose–Einstein statistics. A detailed expose of why photons are said to have zero chemical potential is also proposed.

The foundations of modern probability theory are briefly presented and discussed for both discrete and continuous stochastic variables. Without daring to give a rigorous mathematical construction – but citing different extremely well-written handbooks on the matter – the axiomatic theory of Kolmogorov and the concepts of joint, conditional, and marginal probability are introduced, along with the operations of union and intersection of generic random events. Eventually, Bayes’ formula is put forward with some examples. This will be at the cornerstone of statistical inference methods reported in Chapters 5 and 6.

Our last chapter is devoted to entropy. With this excuse we first present Shannon’s information theory, including the derivation of his entropy, and the enunciations and proofs of the source coding theorem and of the noisy-channel coding theorem. Then, we consider dynamical systems and the production of entropy in chaotic systems, termed Kolmogorov–Sinai entropy. For non-experts or readers who require a memory jog, we make a short recap of statistical mechanics. That is just enough to tie up some knots left untied in Chapter 4, when we developed large deviations theory for independent variables. Here we generalize to correlated variables and make one application to statistical mechanics. In particular, we find out that entropy is a large deviations function, apart from constants. We end with a lightning fast introduction to configurational entropy in disordered complex systems. Just to give a tiny glimpse of … what we do for a living!

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.