Resonance phonon approach to phonon relaxation time and mean free path in one-dimensional nonlinear lattices

We extend a previously proposed resonance phonon approach that is based on the linear response theory. By studying the complex response function in depth, we work out the phonon relaxation time besides the oscillating frequency of the phonons in a few one-dimensional nonlinear lattices. The results in the large wave-number-$k$ regime agree with the expectations of the effective phonon theory. However, in the small-$k$ limit they follow different scaling laws. The phonon mean free path can also be calculated indirectly. It coincides well with that derived from the anharmonic phonon approach. A power-law divergent heat conduction, i.e., the heat conductivity $\kappa$ depends on lattice length $N$ by $\kappa \sim N^{\beta }$ with $\beta >0$, then is supported for the momentum-conserving lattices. Furthermore, this approach can be applied to diatomic lattices. So obtained relaxation time quantitatively agrees with that from the effective phonon theory. As for the mean free path, the resonance phonon approach can detect both the acoustic and the optical branches, whereas the anharmonic phonon approach can only detect a combined branch, i.e., the acoustic branch for small $k$ and the optical branch for large $k$.

Figure 1

The dissipation ${\widehat{\mathrm{\Phi }}}_n^{\prime }\left(\omega \right)$ for various $\omega$ in two momentum-conserving lattices [(a) the $\mathrm{FPU}-\beta$ lattice and (b) the $\mathrm{FPU}-\alpha 2\beta$ lattice] and a momentum-nonconserving lattice [(c) the ${\varphi }^4$ lattice]. The curves from far to near correspond to $\omega =0.153,0.307,0.46,0.614,0.767,0.92,1.23$, and 1.381, respectively. $N=1024,T=0.2$.

Figure 2

The columns from left to right correspond to the $\mathrm{FPU}-\beta$ lattice, the $\mathrm{FPU}-\alpha 2\beta$ lattice, and the ${\varphi }^4$ lattice, respectively. The upper row: the relaxation time ${\tau }_k$ of the space modes from r-ph (symbols) and the corresponding expectations by the e-ph theory (curves). Clear discrepancies can be seen, and the e-ph theory is not applicable to the $\mathrm{FPU}-\alpha 2\beta$ lattice. The lower row: the MFP $l_k$ of the space modes (symbols) and the corresponding expectations by the a-ph approach (curves). The agreement is basically satisfying. However, the a-ph approach is hardly applicable to the high temperature $T=10$ for the ${\varphi }^4$ lattice. The slopes $k^{-1.7}$ in (a) and (d) and $k^{-1.35}$ in (b) and (e) are drawn for reference.

Figure 3

The space modes in a diatomic $\mathrm{FPU}-\beta$ lattice. $N=2048,T=0.2$. (a) and (b) are for the cases that the central 0th particle is a heavy one $(m_0=1.618)$ and a light one $(m_0=1)$, respectively. Two types of oscillations with high and low frequencies can be observed for $k=\pi /32$. The high frequency one decays faster, and the low frequency one decays slower. In the case of $k=\pi /2$, only one frequency mode remains.

Figure 4

The diatomic $\mathrm{FPU}-\beta$ lattice. $N=2048,T=0.2$. The results derived from the r-ph approach are plotted as symbols. The blue squares denote the acoustic phonon branch, and the red triangles denote the optical one. The left and right columns are for the cases that the central 0th particle is a heavy one $(m_0=1.618)$ and a light one $(m_0=1)$, respectively. The first row: the magnitudes $I_k^a$ and $I_k^o$. The negative values mean that when $t=0$ the corresponding oscillation starts not from a crest but from a trough, see, for example, the case of $k=\pi /32$ in Fig. 3. The second row: the relaxation time ${\tau }_k$. The blue dashed and red solid curves are for the acoustic and optical branches, respectively, derived from the e-ph theory. The agreement with the r-ph approach is acceptable. The third row: the phonon MFP $l_k$. The black solid curves are for the a-ph results. The r-ph MFP displays two separated branches, whereas the a-ph MFP displays only a combined branch.