On the kinetic temperature of a one-dimensional crystal on the long-time scale

We investigate the dynamics of the kinetic temperature of a finite one-dimensional harmonic chain, the evolution of which is initiated by a thermal shock. We demonstrate that the kinetic temperature returns arbitrarily close to its initial state (the one immediately following the thermal shock) infinitely many times, and we give an estimate for the time elapsed until the recurrence. This assertion is closely related to the Poincare recurrence theorem and we discuss their relation. To estimate the recurrence time we use its averaging along system's trajectory and provide a rigorous mathematical definition of the mean recurrence time. It turns out that the mean recurrence time exponentially increases with the number of particles in the chain. A connection is established between this problem and the local theorems of large deviations theory. Previous studies have shown that in such a one-dimensional harmonic chain, at times of order N, a thermal echo phenomenon is observed - a sharp increase in the amplitude of kinetic temperature fluctuations. In the present work, we give a rigorous mathematical formulation to this phenomenon and estimate the amplitude of the fluctuations. The research is funded by the Ministry of Science and Higher Education of the Russian Federation within the framework of the Research Center of World-Class Program: Advanced Digital Technologies (agreement & numero;075-15-2020-311 dated 04.20.2022).