Dissipation function in a magnetic field (Review)

The dissipation function is introduced to describe the behavior of the system of harmonic oscillations interacting with the environment (thermostat). This is a quadratic function of generalized velocities, which determines the rate of dissipation of the mechanical energy in the system. It was assumed earlier (Landau, Lifshitz) that the dissipation function can be introduced only in the absence of magnetic field. In the present review based on the author’s studies, it has been shown how the dissipation function can be introduced in the presence of a magnetic field $\mathbf{B}$. In a magnetic field, both dissipative and nondissipative responses arise as a response to perturbation and are expressed in terms of kinetic coefficients. The matrix of nondissipative coefficients can be obtained to determine an additional term formally including it into the equations of motion, which still satisfy the energy conservation law. Then, the dissipative part of the matrix can be considered in exactly the same way as without magnetic field, i.e., it defines the dissipation loss. As examples, the propagation and absorption of ultrasound in a metal or a semiconductor in a magnetic field have been considered using two methods: (i) the method based on the phenomenological theory using the equations of the theory of elasticity and (ii) the method based on the microscopic approach by analyzing and solving the kinetic equation. Both examples are used to illustrate the approach with the dissipation function.