On the fundamental solution of the heat transfer problem in one-dimensional harmonic crystals

Unsteady thermal processes in low-dimensional structures are considered. Understanding the heat transfer at the micro-level is necessary to obtain a link between micro- and macroscopic descriptions of solids. At the macroscopic level, heat propagation is described by the Fourier law. However, at the microscopic level, analytical, numerical and experimental studies show significant deviations from this law. The previously created model of heat transfer at the microlevel, which has a ballistic character, is used in the work. The influence of non-nearest neighbors on the thermal processes in discrete media is studied, as well as the heat distribution in polyatomic lattices is considered. To describe the evolution of the initial thermal perturbation, the analysis of dispersion characteristics and group velocities in a one-dimensional crystal for a diatomic chain with alternating masses or stiffnesses and a monoatomic chain with regard for interaction with second neighbors is carried out. The fundamental solution of the heat propagation problem for the corresponding crystal models is obtained and studied. The fundamental solution allows us to obtain a description of the waves running from a point source and to use it as a basis for the construction of all other solutions. For both chains, the solution consists of two fronts moving one after another with different velocities and intensity characteristics. Quantitative estimations of the thermal wavefront intensity coefficients are given, and the dynamics of changes in speeds and intensities of the waves depending on the parameters of the problem is analyzed. Two mechanisms of evolution of the heat wavefront in one-dimensional discrete systems are revealed. The presented results can be used for correct interpretation of experiments on unsteady ballistic heat transfer in crystals.