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 oscillations amplitude is realized. This phenomenon, further refereed to as the thermal echo, is realized periodically, with the period proportional to the crystal size. The amplitude for each subsequent echo is lower than for the previous one. It is obtained analytically that the time-dependance of the kinetic temperature can be described by an infinite sum of the Bessel functions with multiple indexes. It is also shown that the thermal echo in the thermodynamic limit is described by the Airy function.

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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.

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