Cross-correlations between phonon modes in anharmonic oscillator chains: Role in heat transport

We have computed current-current correlation functions in chains of anharmonic oscillators described by various models (FPU-$\beta$, FPU-$\alpha \beta$, ${\varphi }^4$), considering both the total current and the currents associated with individual phonon modes, which are important in view of the Green-Kubo relation for heat conductivity. Our simulations show that, contrary to the common hypothesis, there are, under some circumstances, significant correlations between neighboring modes. These cross-mode correlations are the dominant contribution to the conductivity in the low anharmonicity regime. The inverse of the timescale over which they are significant, $1/{\tau }_c$, is related to the anharmonicity level in a way similar to the largest Lyapunov exponent, suggesting that the two quantities are related. Cross-mode correlations exist in both anomalous and regular heat-conducting systems although we are unable to observe a transition to the independent-mode regime in the latter case.

Figure 1

(a) Current-current correlation functions $C\left(t\right)$, $C_d\left(t\right)$, and $C_{od}\left(t\right)$ for the FPU-$\beta$ model with $\beta =0.01$. (b) Same data as (a) but in log-log form. (c) Correlation ratio $R\left(t\right)$ for the same system.

Figure 2

(a) Current-current correlation functions $C\left(t\right)$, $C_d\left(t\right)$, and $C_{od}\left(t\right)$ for the FPU-$\beta$ model with $\beta =0.1$. (b) Same data as (a) but in log-log form. (c) Correlation ratio $R\left(t\right)$ for the same system.

Figure 3

(a) Current-current correlation functions $C\left(t\right)$, $C_d\left(t\right)$, and $C_{od}\left(t\right)$ for the FPU-$\beta$ model with $\beta =1$. (b) Same data as (a) but in log-log form. $C_{od}\left(t\right)$ is shown up to ${log}_{10}t=4$ only (when noise starts to dominate), to avoid obscuring the other curves. (c) Correlation ratio $R\left(t\right)$ for the same system.

Figure 4

Current-current correlation functions $C\left(t\right)$ and $C_d\left(t\right)$ scaled by $1/N$ for the FPU-$\beta$ model with $N=256$, 2048, and $16384$ and (a) $\beta =0.01$ and (b) $\beta =0.1$.

Figure 5

Current-current correlation functions $C_{24,24,2048}\left(t\right)$, $C_{3,3,256}\left(t\right)$, $C_{24,+,2048}\left(t\right)$, and $C_{3,+,256}\left(t\right)$ for the FPU-$\beta$ model with (a) $\beta =0.01$ and (b) $\beta =0.1$. In the first case the $C_{k,k,N}\left(t\right)$ curves are found to be different, while the $C_{k,+,N}\left(t\right)$ curves are similar; in the second case all curves are relatively similar.

Figure 6

Current-current correlation functions $C_{8,p,2048}\left(t\right)$ for the FPU-$\beta$ model with $\beta =0.01$ and various values of $p$. Higher values of $p$ lead to an earlier peak, and values closest to 8 lead to a higher maximum.

Figure 7

Lyapunov exponent $\lambda$ and inverse correlation time $1/{\tau }_c$ as a function of $\beta$ for the FPU-$\beta$ model with $T=1$ and $N=2048$. Straight lines correspond to fits to the low- and high-anharmonicity data points. The transition is at $\beta =0.099$ for $\lambda$ and at $\beta =0.73$ for $1/{\tau }_c$. Arrows indicate that the associated data are upper bounds on $1/{\tau }_c$, as transitions were not observed for these simulations.

Figure 8

Current-current correlation functions $C\left(t\right)$, $C_d\left(t\right)$, and $C_{od}\left(t\right)$ for the FPU-$\alpha \beta$ model with (a) $\alpha =\beta =0.01$ and (b) $\alpha =\beta =0.1$.

Figure 9

Current-current correlation functions $C\left(t\right)$, $C_d\left(t\right)$, and $C_{od}\left(t\right)$ [note that $C\left(t\right)$ and $C_{od}\left(t\right)$ are superimposed for almost all times] for the ${\varphi }^4$ model with (a) $\lambda =0.01$ and (b) $\lambda =0.1$. Note that only the $y$ axis is in log form in both cases.