Triggering waves in nonlinear lattices: Quest for anharmonic phonons and corresponding mean-free paths

Guided by a stylized experiment we develop a self-consistent anharmonic phonon concept for nonlinear lattices which allows for explicit “visualization.” The idea uses a small external driving force which excites the front particles in a nonlinear lattice slab and subsequently one monitors the excited wave evolution using molecular dynamics simulations. This allows for a simultaneous, direct determination of the existence of the phonon mean-free path with its corresponding anharmonic phonon wave number as a function of temperature. The concept for the mean-free path is very distinct from known prior approaches: the latter evaluate the mean-free path only indirectly, via using both a scale for for the phonon relaxation time and yet another one for the phonon velocity. Notably, the concept here is neither limited to small lattice nonlinearities nor to small frequencies. The scheme is tested for three strongly nonlinear lattices of timely current interest which either exhibit normal or anomalous heat transport.

Figure 1

Hunting for phonons and MFPs in nonlinear lattices: (a) An illustration of the tuning fork experiment; (b) a schematic sketch of the driving force method in a nonlinear lattice.

Figure 2

The velocity autocorrelation $\langle v_n\left(t\right)v_1\left(0\right)\rangle$ of the FPU-$\beta$ model for $n=1,11,21$. The simulation is carried out on an FPU-$\beta$ lattice with length $N=2048$ at temperature $T=0.2$.

Figure 3

The response function for an FPU-$\beta$ chain with length $N=2048$ at temperature $T=0.2$. (a) A detailed example for ${\chi }_n\left(\omega \right)$ at frequency $\omega =1.505$ as a function of $n$. The upper panel shows the phase ${\varphi }_n$ and the lower panel shows the amplitude $|{\chi }_n|$. The inset shows the principal values of the arguments $\mathrm{Arg}\left[{\chi }_n\right]\in [-\pi ,\pi )$. The principal value jumps discontinuously by $2\pi$ when $-\pi$ is reached. To obtain a continuous varying phase ${\varphi }_n$, as depicted in the upper panel, we shift the arguments by $2\pi$ after each such jump. (b), (c) A comprehensive view of the phase and the amplitude, respectively, for different frequencies $\omega \in (0.0096,2.675)$. The driving frequency $\omega$ increases along the arrow.

Figure 4

(a) The dispersion relation for the FPU-$\beta$ model. The dashed curves are obtained from the renormalized phonon theory using Eq. (22). (b) Corresponding phonon MFPs. The dashed line serves as a guide to the eye for the power-law behavior $\ell \sim k^{-1.70}$.

Figure 5

The response function for an FPU-$\alpha \beta$ chain with length $N=2048$ at temperature $T=0.2$. (a) A detailed example for ${\chi }_n\left(\omega \right)$ at frequency $\omega =0.566$. The upper panel shows the phase ${\varphi }_n$ and the lower panel shows the amplitude $|{\chi }_n|$. The inset shows the principal values of the arguments $\mathrm{Arg}\left[{\chi }_n\right]\in [-\pi ,\pi )$. The principal value jumps discontinuously by $2\pi$ when $-\pi$ is reached. To obtain a continuous varying phase ${\varphi }_n$, as depicted in the upper panel, we shift the arguments by $2\pi$ after each such jump. (b), (c) A comprehensive view of the phase and the amplitude, respectively, for different frequencies $\omega \in (0.0096,2.387)$. The driving frequency $\omega$ increases along the arrow.

Figure 6

(a) The dispersion relation for the FPU-$\alpha \beta$ model at different temperatures. The dashed curves are obtained from (22) with $\alpha \left(T\right)$ set at the sound speed as theoretically derived in Ref. [45]. (b) The effective MFPs.

Figure 7

(a) Phonon dispersion relation for a ${\varphi }^4$ nonlinear lattice at different temperatures. The dashed curves are obtained from renormalized phonon theory [48]. (b) Corresponding anharmonic phonon MFP behavior.