Kinetic Equation for Elementary Excitations in Quantum Systems

Abstract

Kinetic theory deals with macroscopic systems involving a large number of particles. At Boltzmann’s time the kinetic theory described the properties of systems consisting of particles subject to the laws of classical mechanics. The XX-th century quantum physics has developed a new field — quantum theory of solids. This theory studies the properties of macroscopic systems of particles, their motion being described by the laws of quantum mechanics. In the language of quantum mechanics, the states of such macroscopic quantum systems are defined by their energy levels. A weakly excited state of such a system (beyond the ground state), as can be proved, may be described as a superposition of some elementary excitations, characterized by a certain dependence of their energy on the momentum. This dependence ε (p) is called the energy spectrum of the system. The well known Debye crystal theory is based on these concepts, in which the role of elementary excitations is taken by phonons characterized by a phonon spectrum

$$\varepsilon = cp$$

(1)

Elementary excitations have a behaviour similar to that of particles. The application of the laws of statistical physics to these quasiparticles allows to calculate all the thermodynamical values of the system, as it has been done, for instance, in the Debye theory.