Long-time tail of the velocity autocorrelation function in a two-dimensional moderately dense hard-disk fluid

Alder and Wainwright discovered the slow power decay $\sim t^{-d/2}$ ($d$ is dimension) of the velocity autocorrelation function in moderately dense hard-sphere fluids using the event-driven molecular dynamics simulations. In the two-dimensional (2D) case, the diffusion coefficient derived using the time correlation expression in linear response theory shows logarithmic divergence, which is called the “2D long-time-tail problem.” We reexamined this problem to perform a large-scale, long-time simulation with $1\times {10}^6$ hard disks using a modern efficient algorithm and found that the decay of the long tail in moderately dense fluids is slightly faster than the power decay $\left(\sim 1/t\right)$. We also compared our numerical data with the prediction of the self-consistent mode-coupling theory in the long-time limit $\left[\sim 1/\left(t\sqrt{\text{ln}t}\right)\right]$.

Figure 1

(Color online) The VACFs for a typical packing fraction in terms of inverse time are shown. The particle number and packing fractions are fixed at $N=1024$ and $\nu =0.18,0.30,0.45$, respectively. With these parameters, this figure corresponds to Fig. 3 in Ref. [1].

Figure 2

(Color online) The system-size dependence of the VACFs on the packing fraction $\nu =0.30$.

Figure 3

(Color online) The packing fraction dependence of VACFs from a dilute to a moderately dense fluid with a $1\times {10}^6$ hard-disk system is shown. The vertical axis of the VACFs is multiplied by $t/t_0$ in the semilogarithmic plot.

Figure 4

(Color online) The packing fraction dependence of the VACFs from a moderately dense fluid to a solid with a $1\times {10}^6$ hard-disk system is shown. The vertical axis of the VACFs is multiplied by $t/t_0$ in the semilogarithmic plot.

Figure 5

(Color online) The VACFs in a moderately dense hard-disk fluid are compared between numerical simulations and the theoretical predictions using the simple MCT [6] (dashed line) and self-consistent MCT [14, 15] (solid line).