Power law tails of time correlations in a mesoscopic fluid model

In a quenched mesoscopic fluid, modeling transport processes at high densities, we perform computer simulations of the single particle energy autocorrelation function $C_e\left(t\right)$, which is essentially a return probability. This is done to test the predictions for power law tails, obtained from mode coupling theory. We study both off and on-lattice systems in one- and two-dimensions. The predicted long time tail $\sim t^{-d∕2}$ is in excellent agreement with the results of computer simulations. We also account for finite size effects, such that smaller systems are fully covered by the present theory as well.

Figure 1

(Color online) Simulation data for EACF, $C_e\left(t\right)$ vs $t$ at different reduced densities $\rho$ [defined below Eq. (2)] in the two-dimensional random solid for a system of $N=\rho {(L∕r_c)}^2=5\times {10}^4$ particles, compared with theoretical predictions: the dashed lines represent the short time exponential decay, and the solid lines the algebraic LTT $\sim 1∕t$. The higher the density the sooner the LTT is reached.

Figure 2

(Color online) Collapse plot of the LTT’s in Fig. 1 obtained by plotting $\rho C_e\left(t\right)∕C_e\left(0\right)$ in Eq. (5) vs $t^{*}$, compared with the predicted LTT (solid line). Note that the crossover from exponential to algebraic decay is at $t_D=r_c^2∕D_T\left(\rho \right)$.

Figure 3

(Color online) A more sensitive comparison of the collapse plot with the theoretical prediction (solid horizontal line), focusing on relatively short times.

Figure 4

(Color online) The log-log plot shows the EACF vs $t$ in the deterministic simulations at $\rho =5$ for $L∕r_c=20$,40,100, corresponding to $N=2\times {10}^3,8\times {10}^3,5\times {10}^4$ particles. The scattered black circles $(L∕r_c=44)$ show stochastic simulations for comparison. The straight solid line shows the predicted power law tail (8), and the dashed line indicates the short time exponential decay.

Figure 5

(Color online) The log-log plot shows the same deterministic simulations as in Fig. 4, but focuses on the intermediate algebraic tail. The straight line is the LTT in the thermodynamic limit. The ultimate long time decay is again exponential. For the scattered black circles see the caption of Fig. 4.

Figure 6

(Color online) The half logarithmic plot focuses on the ultimate exponential decay in the simulations of Figs. 4, 5, which are finite size effects, and are quantitatively accounted for by the first term in Eq. (9), $(4∕N)\exp [-{(2\pi ∕L)}^2{D}_{T}t]$. The solid line at the top right is again the algebraic LTT for thermodynamically large systems. For the scattered black circles, see the caption of Fig. 4.

Figure 7

(Color online) The plot shows the simulated EACF on a one-dimensional lattice with nearest neighbor interactions after different equilibration times $t_0=5,50,500,{10}^4$ and $5\times {10}^5$, as well as the exact solution (11) for the one-dimensional lattice model.

Figure 8

(Color online) The plot shows the same data for $C_e\left(t\right)$ as in Fig. 7, but focuses on short times. Note that the interval around ${\lambda }_{0}t\simeq 1$ also equilibrates very slowly.

Figure 9

(Color online) The EACF $C_e\left(t\right)$ in the one-dimensional lattice model at different $\rho$ for a system of $N={10}^4$ particles compared with the theoretical predictions: the dashed line is the short time exponential decay, and the straight lines are the algebraic LTT $\sim 1∕\sqrt{t}$ (compare with Fig. 1).

Figure 10

(Color online) Collapse plot of the LTT in the one-dimensional lattice model (compare with Fig. 2).

Figure 11

A more sensitive comparison of the collapse plot of Fig. 10 for short times (compare with Fig. 3) with the theoretical predictions.

Figure 12

A comparison of the simulations with the theoretical predictions for the heat diffusivity in the one-dimensional lattice model for $R_1\left(\rho \right)=D_T\left(\rho \right)∕D_{\infty }\left(\rho \right)=(1+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\rho $}\right.)(1+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2\rho $}\right.)$ with a few simulation data. Note that $\rho$ is an integer. There is excellent agreement.