Equilibration of kinetic temperatures in face-centered cubic lattices

We study thermal equilibration in face-centered cubic lattices with harmonic and anharmonic (Lennard-Jones) interactions. Initial conditions are chosen such that the kinetic temperatures, corresponding to three spatial directions, are different. We show that in the anharmonic case the approach to thermal equilibrium has two time scales. The first time scale is the period of atomic vibration. At times of the order of several atomic periods, the approach to equilibrium is accompanied by decaying high frequency oscillations of the temperatures. The oscillations are described analytically using the harmonic approximation. In particular, the characteristic frequencies of the oscillations are calculated. It is shown that the oscillations decay in time more slowly than expected. The second time scale, presented in the anharmonic case only, depends on the initial temperature of the system. Normalizing time by this scale, we obtain numerically a universal curve describing equilibration in the Lennard-Jones crystal over a wide range of temperatures.

Figure 1

Oscillations of the average kinetic temperature, $T$, in the harmonic fcc lattice with random initial velocities. The analytical solution [solid line, Eq. (14)] and numerical solution of the lattice dynamics equations in the harmonic approximation (circles) are shown.

Figure 2

Redistribution of kinetic energy (temperature) among $x$ and $y$ directions in the harmonic fcc lattice. The analytical solution (12) (solid line), numerical solution of the lattice dynamics equations (3) (circles), and the equilibrium value (15) (dashed line) are shown.

Figure 3

Contributions of the branches of the dispersion relation to oscillations of the average kinetic temperature. The quantities $T_1$ (solid line), $T_2$ (dashed line), and $T_3$ (dash-dotted line), defined by formula (14), are shown.

Figure 4

Functions $|{\mathcal{F}}_j|$, defined by (16) [$|{\mathcal{F}}_1|$ (solid line), $|{\mathcal{F}}_2|$ (dashed line), and $|{\mathcal{F}}_3|$ (dash-dotted line)].

Figure 5

Equilibration of kinetic temperatures $T_{xx},T_{yy}=T_{zz}$ in the Lennard-Jones crystal at short times. The analytical solution (12) (solid line) and simulation results for $v_0/v_d=0.05$ (circles) and 0.25 (black points) are shown. The dashed line shows the equilibrium value, as predicted by the nonequipartition theorem (15).

Figure 6

Equilibration of kinetic temperatures $T_{xx},T_{yy}=T_{zz}$ in the Lennard-Jones crystal at large times. The analytical solution (12) (solid line) and simulation results for $v_0/v_d=0.05$ (circles), 0.1 (asterisks), and 0.25 (black points) are shown. The dashed line shows the equilibrium value, as predicted by the nonequipartition theorem (15).

Figure 7

Temperature dependence of the anharmonic time scale ${\tau }_a$ for $v_0/v_d=0.25,0.5,0.75,1$. Numerical results (circles), cubic approximation [formula (23), solid line], and linear approximation (dashed line) are shown.

Figure 8

Equilibration of kinetic temperatures in the Lennard-Jones crystal at different temperatures $T_0$ [$v_0/v_d=0.25$ (black circles), $v_0/v_d=0.5$ (blue crosses), $v_0/v_d=0.75$ (dark green asterisks), and $v_0/v_d=1$ (red circles)]. The dashed line shows the equilibrium value, as predicted by the harmonic approximation (15).

Figure 9

Equilibration of kinetic temperatures in the Lennard-Jones crystal at different $T_0$ [$v_0/v_d=0.25$ (black circles), $v_0/v_d=0.5$ (blue crosses), and $v_0/v_d=0.75$ (dark green asterisks)]. The dashed lines correspond to linear approximations at large times.