Transient diffusion and thermal processes in a finite one-dimensional harmonic crystal.

In this paper, an instant homogeneous thermal perturbation in the periodic one-dimensional harmonic crystal is studied. The exact solution for thermal and diffusive characteristics is obtained, namely, particle velocity dispersion (kinetic temperature) and particle displacement dispersion. It is found that thermal and diffusion processes demonstrate a quasi-periodic recurrence. The recurrence interval is equal to the time it takes the sound wave to travel the half-length of the crystal. The 'thermal echo' (sharp peaks in kinetic temperature) occurs in the system with the specified periodicity. Diffusion characteristics reveal large-scale time changes with a nearly complete return to the initial state at each quasi-period. It is also shown that the spatial mean squared displacements of particles are significantly different from the ensemble mean squared displacements.

Transition to thermal equilibrium in a crystal subjected to instantaneous deformation.

An adiabatic transition between two equilibrium states corresponding to different stiffnesses in an infinite chain of particles is studied. Initially, the particles have random displacements and random velocities corresponding to uniform initial temperature distributions. An instantaneous change in the parameters of the chain initiates a transitional process. Analytical expressions for the chain temperature as a function of time are obtained from statistical analysis of the dynamic equations. It is shown that the transition process is oscillatory and that the temperature converges non-monotonically to a new equilibrium state, in accordance with what is usually unexpected for thermal processes. The analytical results are supplemented by numerical simulations.

Thermal echo in a finite one-dimensional harmonic crystal.

An instant homogeneous thermal perturbation in the finite harmonic one-dimensional crystal is studied. Previously it was shown that for the same problem in the infinite crystal the kinetic temperature oscillates with decreasing amplitude described by the Bessel function of the first kind. In the present paper it is shown that in the finite crystal this behavior is observed only until a certain period of time when a sharp increase of the oscillation amplitude is realized. This phenomenon, further referred to as the thermal echo, occurs periodically, with the period proportional to the crystal length. The amplitude for each subsequent echo is lower than for the previous one. It is obtained analytically that the time-dependence of the kinetic temperature can be described by an infinite sum of the Bessel functions with multiple indices. It is also shown that the thermal echo in the thermodynamic limit is described by the Airy function.