Equilibrium time-correlation functions for one-dimensional hard-point systems

As recently proposed, the long-time behavior of equilibrium time-correlation functions for one-dimensional systems are expected to be captured by a nonlinear extension of fluctuating hydrodynamics. We outline the predictions from the theory aimed at the comparison with molecular dynamics. We report on numerical simulations of a fluid with a hard-shoulder potential and of a hard-point gas with alternating masses. These models have in common that the collision time is zero and their dynamics amounts to iterating collision by collision. The theory is well confirmed, with the twist that the nonuniversal coefficients are still changing at longest accessible times.

Figure 1

MD simulation of an equal mass chain with shoulder potential as defined in Eq. (2) and parameters $N=4096$, $p=1.2$, $\beta =2$, at $t=1024$. (a) Diagonal matrix entries, $S_{\alpha \alpha }^{♯}(j,t)$, of the two-point correlations. The gray vertical lines show the sound speed predicted from theory. The tails of the sound peaks reappear on the opposite side due to periodic boundary conditions. (b) Rescaled heat, and (c) right sound peak. The theoretical scaling exponents are used and $\lambda$ is fitted numerically to minimize the $L^1$ distance between simulation and prediction. The dashed orange curve is the predicted $\frac{5}{3}$-Lévy distribution $f_{\mathrm{L},5/3}$ and the dashed red curve shows $f_{\mathrm{KPZ}}$.

Figure 2

Difference between, respectively, the heat and right sound peaks obtained from the MD simulation with shoulder potential and the theoretical prediction at optimal $\lambda$, as listed in Table 1. The notches around $\left|x\right|\simeq 12$ in (a) are due to feedback from the sound modes.

Figure 3

Difference between, respectively, the heat and right sound peaks obtained from the MD simulation with alternating masses $m_0=1$, $m_1=3$, and the theoretical prediction at optimal $\lambda$. Each $\lambda$ is fitted numerically to minimize the $L^1$ distance, see Table 2. Note the notches at $\left|x\right|\simeq 16$ in (a) resulting from feedback of the sound modes.

Figure 4

Difference between the, respectively, rescaled heat and right sound peaks obtained from the MD simulation with maximal distance $a=1$ and alternating masses, and the theoretical prediction at optimal $\lambda$. Note that the exponents and asymptotic functions are different from the previous cases. The fitted $\lambda$ values are provided in Table 3. In the top row, the feedback from the sound modes to the heat mode is clearly visible.

Figure 5

Logarithmic plot of the heat and right sound peaks for all three models, at $t=1024$. The dashed orange curve in (a) and (b) is the $f_{\mathrm{L},5/3}$ and the dashed brown curve in (c) the $f_{\mathrm{L},3/2}$. The dashed red curve in (d) and (e) shows $f_{\mathrm{KPZ}}$, and the magenta dashed curve in (f) is the Gaussian $f_{\mathrm{G}}$.