Finite-size effects on current correlation functions

We study why the calculation of current correlation functions (CCFs) still suffers from finite-size effects even when the periodic boundary condition is taken. Two important one-dimensional, momentum-conserving systems are investigated as examples. Intriguingly, it is found that the state of a system recurs in the sense of microcanonical ensemble average, and such recurrence may result in oscillations in CCFs. Meanwhile, we find that the sound mode collisions induce an extra time decay in a current so that its correlation function decays faster (slower) in a smaller (larger) system. Based on these two unveiled mechanisms, a procedure for correctly evaluating the decay rate of a CCF is proposed, with which our analysis suggests that the global energy CCF decays as $\sim t^{-\frac{2}{3}}$ in the diatomic hard-core gas model and in a manner close to $\sim t^{-\frac{1}{2}}$ in the Fermi-Pasta-Ulam-$\beta$ model.

Figure 1

The energy current correlation function for the gas model (a) and the FPU-$\beta$ model (b). The period of oscillations observed in (a) is measured to be $\frac{L}{v_s}$ ($v_s$ is the sound speed of the system).

Figure 2

The spatiotemporal correlation function of local energy (heat) currents for the 1D gas model (a) and the FPU-$\beta$ model (b). The system size is $L=512$ in both cases.

Figure 3

The comparison between the state correlation function $R\left(t\right)$ (blue solid line) and the energy current correlation function $C_{JJ}\left(t\right)$ (red dashed line) for the gas mode (a) and the FPU-$\beta$ model (b). In order to have a close comparison, $C_{JJ}\left(t\right)$ is shifted down by a factor of $\frac{0.3}{N}$. The system size adopted is $L=512$ in both panels.

Figure 4

(a) The comparison of the rescaled energy CCF, $t^{{\gamma }^{*}}{C}_{JJ}\left(t\right)$, for the gas model with ${\gamma }^{*}=\frac{2}{3}$ (orange dashed line) and ${\gamma }^{*}=\frac{3}{5}$ (blue dash-dotted line). The black horizontal line is plotted for reference. The system size is $L=65536$. (b) The same as (a) but for the FPU-$\beta$ model with ${\gamma }^{*}=\frac{3}{5}$ and ${\gamma }^{*}=\frac{1}{2}$ instead. The system size is $L=131072$.