GENERAL PROPERTIES OF MANY-PARTICLE SYSTEMS AT LOW TEMPERATURES

I. Elementary Excitations. The Energy Spectrum and Properties of Liquid He4 at Low Temperatures

1.1. Introduction. Quasi-particles. Statistical physics studies the behavior of systems consisting of a very large number of particles. In the last analysis, the macroscopic properties of liquids, gases and solids are due to microscopic interactions between the particles making up the system. Obviously, a complete solution of the problem, involving determination of the behavior of each individual particle, is out of the question. Fortunately, however, the overall macroscopic characteristics are determined only by certain average properties of the system.

To be explicit, we now consider some thermodynamic properties. The macroscopic state of a system is specified by giving three independent thermodynamic variables, e.g., the pressure P, the temperature T, and the average number N of particles in the system. From a quantum-mechanical point of view, a closed system of N particles is characterized by its energy levels En. Suppose that from the system we single out a volume (subsystem) which can still be regarded as macroscopic. The number of particles in such a subsystem is still very large, whereas the interaction forces between particles act at distances whose order of magnitude is that of atomic dimensions. Therefore, apart from boundary effects, we can regard the subsystem itself as closed, and characterized by certain energy levels (for a given number of particles). Since the subsystem actually interacts with other parts of the closed system, it does not have a fixed energy and a fixed number of particles, and, in fact, it has a nonzero probability of occupying any energy state.

As is familiar from statistical physics (see e.g., L8), the microscopic derivation of thermodynamic formulas is based on the Gibbs distribution, which gives the following probability of finding the subsystem in the energy state with a number of particles equal to N:

(1.1) [MATHEMATICAL EXPRESSION OMITTED]

In this formula, T denotes the absolute temperature, μ the chemical potential, and Z a normalization factor which is determined from the condition

(1.2) [MATHEMATICAL EXPRESSION OMITTED]

According to (1.1), we have

(1.3) [MATHEMATICAL EXPRESSION OMITTED]

The quantity Z is called the grand partition function. If the energy levels EnN are known, the partition function can be calculated. This immediately determines the thermodynamic functions as well, since the formula

(1.4) Ω = -T ln Z

relates the quantity Z to the thermodynamic potential Ω (involving the variables V, T and μ).

Obviously, the simplest use that can be made of these formulas is to calculate the thermodynamic functions of ideal gases, since in this case the energy is just the sum of the energies of the separate particles. However, in general, it is impossible to determine the energy levels of a system consisting of a large number of interacting particles. Therefore, so far, interactions between particles in quantum statistics have been successfully taken into account only when the interactions are sufficiently weak, and perturbation-theory calculations of thermodynamic quantities have been carried out only to the first or second approximations. In the majority of physical problems, where the interaction is far from small, an approach based on the direct use of formulas (1.1)–(1.4) is unrealistic.

The case of very low temperatures is somewhat exceptional. As T -> 0, the important energy levels in the partition function are the weakly excited states, whose energies differ only very little from the energy of the ground state. The character of the energy spectrum of the system in this region of energies can be ascertained in some detail, by using very general considerations which are valid regardless of the magnitude and specific features of the interaction between the particles.

As an example illustrating the subsequent discussion, consider the excitation of lattice vibrations in a crystal. As long as the vibrations are small, we can regard the lattice as a set of coupled harmonic oscillators. Introducing normal coordinates, we obtain a system of 3N linear oscillators with characteristic frequencies ωi(N is the number of atoms). According to quantum mechanics, the energy spectrum of such a system is given by the formula

[MATHEMATICAL EXPRESSION OMITTED]

where the ni are arbitrary nonnegative integers (including zero), and various sets of numbers ni correspond to various energy levels of the system. The lattice vibrations can be described as a superposition of monochromatic plane waves, propagating in the crystal. Each wave is characterized by a wave vector and a frequency, and also by an index s specifying the type of wave involved. Because of the possibility that various types of waves can propagate in the crystal, the frequency to, regarded as a function of the wave vector k, is not a single-valued function, but rather consists of several branches ωs(k), where the total number of branches equals 3r (r is the number of atoms belonging to one unit cell of the crystal). For small momenta, three of these branches (the acoustic branches) are characterized by the fact that the frequency depends linearly on the wave vector:

ωs(k) = us(θ, φ)|k|.

For other momenta, the curve ωs(k) begins with some finite value for k = 0, and depends weakly on k in the region of small wave numbers.

From a knowledge of the frequency spectrum, the energy levels and the matrix elements of the displacements of the atoms of the lattice (the coordinates of the oscillators), we can calculate completely, at least in principle, both the thermodynamic and the kinetic characteristics of the vibrating lattice. However, instead of the model of coupled oscillators, it turns out in practice to be very convenient to use another, equivalent model. This model can be obtained by applying the quantum-mechanical correspondence principle, which states that every plane wave corresponds to a set of moving "particles," with momentum determined by the wave vector k and energy determined by the frequency ωs(k). Thus, an excited state of the lattice can be thought of as an aggregate of such "particles" (called phonons), moving freely in the volume occupied by the crystal. This leads to an expression for the energy levels of the system which is analogous to that for an ideal gas. In fact, ni can be interpreted as the number of...