D. Pines in his Editor's Foreword to the important series “Frontiers in Physics, a Set of Lectures” of the sixties and seventies of the past century (W. A. Benjamin, Inc.) was suggesting as a possible solution to “the problem of communicating in a coherent fashion the recent developments in the most exciting and active fields of physics” what he called “an informal monograph to connote the fact that it represents an intermediate step between lecture notes and formal monographs.”

Our aim in writing this book has been to provide a coherent presentation of different topics, emphasizing those concepts which underlie recent applications of statistical mechanics to condensed matter and many-body systems, both classical and quantum. Our goal has been indeed to reach an up to date version of the book Statistical Mechanics. A Set of Lectures by R. P. Feynman, one of the most important monographs of the series mentioned above. We felt, however, that it would have been impossible to give to a student the full flavor of the recent topics without putting them in the classical context as a continuous evolution. For this reason we introduced the basic concepts of thermodynamics and statistical mechanics. We have also concisely covered topics that typically can be found in advanced books on many-body theory, where usually the apparatus of quantum field theory is used.

In our book we have kept the technical apparatus at the level of the density matrix with the exception of the last four chapters. Up to Chapter 17 no particular prerequisite is needed except for standard courses in Classical and Quantum Mechanics. Chapter 18 provides an introduction to statistical quantum field theory, which is used in the last chapters. Chapters 20 and 21 cover topics which, although covered in recent monographs, are not commonly found in classical many-body books.

In our book then the student will find a bridge from thermodynamics and statistical mechanics towards advanced many-body theory and its applications. In our attempt to give a coherent account of several topics of condensed matter physics, we have at the same time preserved the personal point of view of the notes of our courses. Our bibliography is for this reason far from complete and the presentation of some topics is somewhat informal and partial. Many important contributions and fundamental references have been left out.

According to the second law of thermodynamics, the evolution towards an equilibrium state corresponds to the increase of the entropy of a thermally isolated system, or, depending on the external conditions, by the decrease of an appropriate thermodynamic potential. Statistical mechanics must explain the origin of such a law starting from the laws which control the motion of the microscopic constituents of matter. At a microscopic level, however, all the basic laws are symmetric with respect to time reversal. This suggests that a purely mechanical derivation of the second law of thermodynamics is conceptually impossible and a more subtle reasoning, based on probability considerations, is necessary to explain irreversibility and the thermodynamics of a system in terms of the motion of its elementary constituents. This has been clearly explained by Boltzmann after several years of intense thinking. Towards the end of his life, Boltzmann published (in 1896) Lectures on Gas Theory (Boltzmann (1964)). This is the topic of this chapter and the next.

The birth of kinetic theory

Kinetic theory makes its first appearance in the book Hydrodynamics published by Daniel Bernoulli in 1738 (Bernoulli and Bernoulli (2005)). The assessment of the kinetic approach of Bernoulli was undertaken by Clausius (1857). Bernoulli's argument explains the pressure exercised by a gas on the walls of the container in terms of the collisions of its constituent molecules. Let us imagine a molecule approaching one of the walls of the gas container as shown in Fig. 2.1. We assume elastic scattering and the wall perpendicular to the x-axis. Let vx 0 be the component along the x-axis of the initial velocity. Since upon bouncing on thewall the molecule reverses the sign of its perpendicular velocity component, the transferred momentum to the wall is 2mvx, m being the mass of the molecule. The transferred momentum per unit time is the force exercised by the molecule on the wall during a single collision. The total force on the wall is the cumulative effect of the collisions occurring in unit time. The number of collisions is the number of molecules which during the unit time, dt, will collide with the wall, i.e. those which are vxdt far away from the wall.

As discussed in the previous chapter, the phenomenological theory of critical phenomena finds a sound basis in the logical sequence of universality, scaling, relevant and irrelevant variables. The difficult problem of a large number of degrees of freedom strongly correlated within the coherence distance, which diverges at criticality, is thereby, in principle, reduced to the determination of the homogeneous form of the relevant response functions and of the evaluation of a few critical indices related to the relevant parameters, whereas the details specifying each system do not matter.

In order to give a microscopic foundation to universality and to the scaling theory, an exact transformation is needed, which changes into a scaling transformation asymptotically near the critical point. This is the renormalization group (RG) approach, recently presented in a huge number of reviewpapers and books.Historically, the first comprehensive overview was given by the sixth volume of the series edited by Domb and Green (1976) and by the classic books by Patashinskij and Pokrovskij (1979) and by Ma (1976).

Along with scaling theory, the field-theoretic renormalization group (RG) approach (Gell- Mann and Low (1954); Bogoliubov and Shirkov (1959); Bonch-Bruevich and Tyablikov (1962)) was introduced in critical phenomena (Di Castro and Jona-Lasinio (1969)) by showing how the RG equations generalize the universality relations in the sense that they relate one model system to another by varying the coupling and suitably rescaling the other variables and the correlation functions. At the same time a detailed analysis of the pertubative diagrammatic structure of the correlation functions together with the use of Ward identities yielded a renormalized theory (Migdal (1969); Polyakov (1968, 1969)). By iterating the RG transformation the coupling disappears from the equations near to the critical point, the homogeneous form of the static order-parameter correlation function is obtained and a microscopic definition of the critical indices is given. The scaling picture then acquires a theoretical basis.

Wilson (Wilson (1971a, b); Wilson and Kogut (1974)) gave a great contribution to the understanding of the physics underlying the RG procedure and to the explicit calculation of the critical indices. He proposed a simple mathematical realization of the other very physical idea of universality due to Kadanoff of grouping together degrees of freedom on the scale s associated with larger and larger cells.

For pure fermionic systems, and in the absence of symmetry breaking (e.g. magnetism, superconductivity, …), two types of metallic phases are well established: (i) the “normal” Fermi liquid phase in d = 3 presented in Chapters 12 and 19 via the response and the Green functions techniques; and (ii) the “anomalous” Luttinger liquid phase in d = 1 originally formulated with the bosonization technique by which the fermionic operators are represented in terms of “bosonic” density operators (Luther and Peschel (1974); Haldane (1981)) with the exact solution (Mattis and Lieb (1965)) of the Tomonaga–Luttinger model (Tomonaga (1950); Luttinger (1963)) discussed inmany reviews and books (see for instance the book by Giamarchi (2004)).

Here, to stress similarities and differences between the Fermi liquid and the Luttinger liquid, we will follow the alternative route of response and Green functions techniques (Dzyaloshinskii and Larkin (1973), Di Castro and Metzner (1991), Metzner et al. (1998)) and add a few comments on the renormalization group approach (Metzner and Di Castro (1993)).

As we have seen in Chapters 12 and 19, the Fermi liquid is described asymptotically by the free low-lying single-particle excitations, i.e., the quasiparticles with a zero-kelvin discontinuous occupation number in momentum space nk (see Eq. (18.54)) at a well defined Fermi surface (k = k F):

where here k F> and k F< indicate the limit k → kF from above and below. This discontinuity still marks the Fermi surface in the interacting system. The finite reduction of the single-particle spectral weight zkF with respect to the Fermi gas (zkF = 1) is given by the finite (i.e. non-critical) “wavefunction renormalization” (see Eq. (18.94)). The presence of the discontinuity at the Fermi surface, together with the Pauli principle, compel the inverse quasiparticle lifetime to be τ−1 ≈ max(T2, ϵ2), where _ is the deviation of the energy of the quasiparticle from the Fermi energy. The energy uncertainty ħτ−1 due to the finite lifetime of a quasiparticle near the Fermi surface is small compared to its energy ϵ, and the quasiparticle concept is well defined. All the momentum transferring scattering processes become asymptotically ineffective, as derived phenomenologically in Chapter 12 and confirmed microscopically in Chapter 19.