Numerical extraction of sound velocities of energy transfer based on solitons in Fermi-Pasta-Ulam chains

According to the Boltzmann distribution assumption of solitons, in thermal equilibrium, there is the most probable soliton whose average kinetic energy per site equals the thermal energy $k_{B}T$. Based on the momentum excitation method, the soliton can be numerically excited in the static Fermi-Pasta-Ulam (FPU) chains. By associating the excited soliton with the corresponding most probable soliton, the temperature dependence of the velocity of solitons in thermal equilibrium can be numerically evaluated. The results agree very well with the temperature dependence of the sound velocity of energy transfer. This confirms that solitons are promising candidates for energy carriers in FPU chains. Moreover, the validity of the Boltzmann distribution assumption of solitons in FPU chains is also confirmed. This work sheds light on how to numerically (even experimentally) investigate solitons in thermal equilibrium.

Figure 1

Snapshot of the compressional soliton in the FPU $\beta$ chain at time 10 000. The initial excitation momentum is $p=0.8$.

Figure 2

Velocity of solitons vs ${\epsilon }_k/k_B$ for the one-dimensional FPU $\beta$ chain. The symbols correspond to the numerical results. The solid line corresponds to the prediction of NFH for the dependence of the sound velocity on the temperature $T$.

Figure 3

Snapshot of the solitons in the quartic chain at time 2000. The initial excitation momentum is $p=0.1$.

Figure 4

Velocity of solitons vs ${\epsilon }_k/k_B$ for the one-dimensional quartic chain. The symbols correspond to the numerical results. The solid line corresponds to the prediction of NFH for the dependence of the sound velocity on the temperature $T$.

Figure 5

Snapshot of the solitons in the FPU $\alpha \beta$ chain at time 10 000. The initial excitation momentum is $p=2.6$. The inset shows a close-up of the solitons.

Figure 6

Velocity of solitons vs ${\epsilon }_k/k_B$ for the one-dimensional FPU $\alpha \beta$ chain. The triangles correspond to the numerical results of the compressional solitons. The squares correspond to the numerical results of the rarefaction solitons. The dashed line and the solid line represent the corresponding analytical results of the dependences of the velocities on the temperature $T$.