Temperature oscillations in harmonic triangular lattice with random initial velocities

Transition to thermal equilibrium in a two-dimensional harmonic triangular lattice with nearest neighbor interactions is investigated. Initial conditions, typical for molecular dynamics simulations, are considered. Initially, particles have uncorrelated random velocities, corresponding to initial kinetic temperature of the system, and zero displacements. These initial conditions can be realized by heating of the system by an ultrafast laser pulse. In this case, the kinetic temperature of the system oscillates. The oscillations are caused by the redistribution of energy between kinetic and potential parts. At large times, energies equilibrate and temperature tends to the equilibrium value equal to a half of the initial temperature. In our previous works, an integral exactly describing this transient thermal process has been derived. The integrand depends on the dispersion relation for the lattice. The integral contains large parameter, notably time. In the present work, we investigate large time behavior of the kinetic temperature. Simple asymptotic expression for deviation of temperature from the steady state value is derived. The expression contains three harmonics with different frequencies and amplitudes. Group velocities corresponding to these frequencies are equal to zero. Two frequencies are close and therefore beats of kinetic temperature are observed. Amplitude of deviation of temperature from the steady state value decreases inversely proportional to time. It is shown that the asymptotic formula has reasonable accuracy even at small times of order of one period of atomic vibrations.