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.

Thermodynamics is a phenomenological theory based on empirical observations. The thermodynamic laws are a coherent account of the experimental analysis and provide a universally valid description of the behavior of macroscopic systems without referring to their detailed structure. In this chapter we introduce thermodynamics as a prerequisite scenario for statistical mechanics and we do not give an extended exposition of the theory, but only recall the basic concepts as an introduction to the statistical approach.

Equilibrium states and the empirical temperature

A thermodynamic system is any portion of matter, solid, liquid, mixtures, …, which can be described in terms of a small number of parameters, the macroscopic thermodynamic variables, as for example the pressure P and the volume V for a fluid or the magnetic moment and the magnetic field for a paramagnet. These variables are called extensive or intensive, depending whether their value does or does not depend on the amount of matter present in the system. Energy, volume and magnetization are extensive variables, whereas pressure, temperature and magnetic field are intensive.

Thermodynamics deals with macroscopic properties of macro-systems at equilibrium. A thermodynamic equilibrium state must then be characterized as a macroscopic phenomenon defined via macroscopic variables. A system is in equilibrium if the values of the observables we take into consideration and of the parameters describing its state do not change with time. The thermodynamic equilibrium is never fully static. The characterization of a thermodynamic state depends on the observation time, usually at least of the order of fractions of a second, which in any case must always be much longer than any characteristic time scale of the molecular motion (∼10−11 s).

We use a simple example of common experience to illustrate this point. Soon after hot water is poured into a container, it appears to be at rest. If we ask whether the water is in a state of equilibrium, the answer depends obviously on the observation time and on the sensitivity of the measurement. Within the interval of time of a few seconds we do not feel changes in the temperature and in the volume of the water, which during this observation time can be considered at equilibrium by itself.

This is a book for students who have some familiarity with general physics and the calculus to learn what thermodynamics and statistical mechanics is all about, starting with the fact that you don't have to know what a system is doing, just how many things it could be doing, to get precise values of its thermodynamic variables, particularly its temperature and pressure. That insight is discussed and explained in the first chapter, along with some simple observations about the peculiar behavior of very large numbers. It uses the perfect gas to illustrate its points.

Chapter 2 looks into the behavior of thermodynamic variables and gives a lesson in partial derivatives, the essential language of thermodynamics. It also considers the various ways of expanding an ideal gas. Chapter 3 deals with the states of matter and the phase transitions between them, and also covers temperature scales, the ideal gas equation of state and the behavior of the atmosphere.

Chapter 4 considers the origin and meaning of the first and second laws of thermodynamics and goes on to discuss Carnot engines and refrigerators and the notion of reversibility. Chapter 5 addresses the probability of finding a system in a particular state and goes on to deal with the partition function and the density of states.

From atoms to thermodynamics

Imagine a box, a cube 10 cm on each edge, with 1022 atoms of helium gas in it. The atoms share among them some total energy U; say, 2 × 106 ergs, which cannot change because the box is isolated from the rest of the world. Inside the box the atoms fly around, banging into each other or the walls, exchanging energy and momentum. If there is only one atom in the box, and we know how it started out, we might imagine being able to calculate its precise trajectory for a while, predicting just where it would end up at some later time. If there are twenty atoms, the same job becomes horribly more complicated. With 1022 atoms it is obviously hopeless. Moreover, according to the laws of quantum mechanics, it would not be possible even in principle. If we knew precisely where the atoms were at some time, we could have no idea of how fast they were moving, according to the uncertainty principle. Obviously, a very short time after we start things off, there is not much we can say about what's going on inside the box.

Nevertheless, it is possible to make some very precise statements about the properties of the gas in the box, especially if we allow some time to pass after we start it off. For example, the gas will have some pressure, P, and some temperature, T, and, given the information we already have, these can be predicted with extreme accuracy and confidence. Temperature and pressure are macroscopic or thermodynamic quantities. The problem before us in this section is to describe the connection between these (predictable) thermodynamic quantities and the (unpredictable) microscopic quantities that somehow give rise to them.