Molecular dynamics study of relaxons in the Fermi-Pasta-Ulam-$\beta$ model

We present a method to compute the phonon scattering matrix of a given lattice that uses molecular dynamics simulations and is therefore able to capture perturbation terms in the Hamiltonian to all orders. This allows the computation of the relaxons' eigenmodes associated with the system as well as their properties with regard to the flow of heat in the system. Applying this method to the one-dimensional Fermi-Pasta-Ulam-$\beta$ ($\mathrm{FPU}\beta$) lattice, we show that there is an approximate one-to-one mapping from the phonon eigenmodes to the relaxon eigenmodes. For every temperature under study, which span the transition from the weakly stochastic to the strongly stochastic regime, we find that a single relaxon mode with a very long lifetime plays a very important role in the heat flow dynamics. Other relaxons have relaxation times similar to that of the phonons. These conclusions are fully consistent through the various anharmonic regimes associated with the $\mathrm{FPU}\beta$ model.

Figure 1

Phonon damping functions ${\mathrm{\Gamma }}_{kp}\left(s\right)$ at $T=1$ as function of time delay for (a) a diagonal case ${\mathrm{\Gamma }}_{5,5}$ and (b) a nondiagonal case ${\mathrm{\Gamma }}_{18,25}$.

Figure 2

Index $k_{\text{max}}^{\alpha }$ denoting the maximum component of eigenvectors ${\theta }^{\alpha }$ as a function of the relaxon index $\alpha$ for even and odd relaxons at $T=0.01$. The slightly sublinear trend illustrates that relaxons generally tend to be similar to the phonon mode with $k=\alpha$. The particular mode ${\alpha }^o=14$ that stands out is discussed in Sec. 4d.

Figure 3

(a) Relaxon eigenvectors ${\theta }_k^7$ for (bottom) $T=1$ and (top, shifted) $T=0.01$. The sharper peaks in the first case indicate that the relaxon is more phononlike at high temperatures. (b) Eigenvectors ${\theta }_k^3$ (bottom) for an even mode and (top, shifted) for an odd mode for $T=0.01$. Although the odd mode has a lower relaxation time, the even mode is significantly less phononlike. (c) Eigenvectors of odd modes (top) ${\theta }_k^{38}$, (middle) ${\theta }_k^{43}$, and (bottom) ${\theta }_k^{48}$. As the relaxation times decrease, the eigenvectors are formed by fewer and fewer modes. Each datum is presented as a ratio of the adimensional population ${\tilde{n}}_k$.

Figure 4

Relaxation times associated with relaxons modes for $T=0.01$. Full symbols correspond to odd relaxons and open symbols to even relaxons. Trend lines fitted to the first ten modes in each case display power-law behaviors with (odd, solid line) ${\tau }_{\text{odd}}\sim {\alpha }^{-1.97}$ and (even, dashed line) ${\tau }_{\text{even}}\sim {\alpha }^{-1.84}$.

Figure 5

Relaxation times for relaxons and phonons (diagonal and including cross correlations, respectively) in lattices at $T=0.01$. Only odd relaxons are shown as even relaxons do not contribute to the heat flow. Phonons modes with negative wave vectors possess the same relaxation times and mode conductivities as modes with positive wave vectors, and are therefore not shown. The abscissa labels both the relaxon index $\alpha$ and the reduced phonon wave vector $k$. In the case of the total phonon relaxation times, full symbols indicate that the mode displays negative (resistive) correlations.

Figure 6

Relaxation times for relaxons and phonons (single mode or including cross correlations) at $T=0.1$. Only the first relaxon mode displays a longer lifetime than the (total) phonon relaxation time. Given the very high conductivity from low-index modes, this balances the total conductivity obtained with the two approaches.

Figure 7

Relaxation times for relaxons and phonon (single mode or including cross correlations) at $T=1$. Only a single relaxon has a relaxation time longer than the corresponding phonon mode. In the case of this fairly small system, this illustrates that a single relaxon carries most of the heat in the system.

Figure 8

Velocities of relaxons with odd eigenvectors as a function of relaxon index $\alpha$ for temperature $T=0.01$, 0.1, and 1.

Figure 9

Odd relaxon eigenvectors (a) ${\theta }_{13}^k$ and (b) ${\theta }_{14}^k$ for $T=0.01$. The former shows out-of-phase elements that wield very low velocity while the latter displays in-phase contribution from a large number of modes producing a high velocity.