An analytical description of transient thermal processes in harmonic crystals

We consider two transient thermal processes in uniformly heated harmonic crystals: (i) equalibration of kinetic and potential energies and (ii) redistribution of the kinetic energy among the spatial directions. Equations describing these two processes in two-dimensional and three-dimensional crystals are derived. Analytical solutions of these equations for the square and triangular lattices are obtained. It is shown that the characteristic time of the transient processes is of the order of ten periods of atomic vibrations. The difference between the kinetic and potential energies oscillates in time. For the triangular lattice, amplitude of the oscillations decays inversely proportional to time, while for the square lattice it decays inversely proportional to the square root of time. In general, there is no equipartition of the kinetic energy among spatial directions, i.e. the kinetic temperature demonstrates tensor properties. In addition, the covariance of velocities of different particles is nonzero even at the steady state. The analytical results are supported by numerical simulations. It is also shown that the obtained solutions accurately describe the transient thermal processes in weakly nonlinear crystals at short times.