Transition to thermal equilibrium in a uniformly heated 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.

An infinite harmonic triangular lattice with a vacancy under imposed volumetric strain is investigated. Displacement field around the vacancy is considered. In our previous works, an exact displacement field is obtained in the integral form. The integral contains large parameter, notably particle index proportional to distance from the vacancy. In the present paper, behavior of the displacement field far from the vacancy is considered. Asymptotic expansion of the exact solution yields simple expression for the displacement field. The expression has reasonable accuracy for all lattice nodes and it rapidly converges to the exact solution with increasing distance from the vacancy. Strain concentration factor, defined as the ratio of the maximal deformation of the bonds adjacent to the vacancy to the deformations of bonds at infinity, is calculated. It is shown that the asymptotic formula predicts the strain concentration factor with 4 % accuracy. The results are compared with predictions of continuum elasticity theory. In continuum formulation, the lattice with vacancy is modelled by an elastic plate with circular hole. It is shown that the continuum displacement field has the same form as the asymptotic expression for the discrete displacement field. Comparison of discrete and continuum results yields an effective diameter of the vacancy. The comparison also shows that influence of vacancies on strength of the lattice cannot be accurately modeled using continuum theory.

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