Inhomogeneous Distribution of Thermal Characteristics in Harmonic Crystal

In this paper, we consider a model of a uniform harmonic chain of particles for the analysis of non-stationary thermal effects in an ideal crystal system. The exact solution for the particle system is presented and the temperature is calculated as a measure of the average kinetic energy of the particles. The corresponding energy averaging is performed over the initial distribution of the displacements and velocities of the particles, provided that they obey the Boltzmann principle. Simple analytical formulae are presented for all energy derivatives with respect to time at the initial time and for the first derivative with respect to the number of particles. Over a small time interval, the temperature was shown to depend monotonically on the number of particles. This means that the non-uniformity of thermal characteristics distribution, i.e. dependence on the number of particles, occurs in the system without additional assumptions about the structure of the initial conditions on a macroscopic scale. The obtained formula for the distribution of kinetic energy is presented through Bessel functions. The functional dependence on the number of particles was shown to appear in the index of Bessel functions, and the parity of the number of particles affects the temperature distribution. The distribution of the kinetic energy for a large time was asymptotically analyzed as well.