Fourier heat conduction as a strong kinetic effect in one-dimensional hard-core gases

For a one-dimensional (1D) momentum conserving system, intensive studies have shown that generally its heat current autocorrelation function (HCAF) tends to decay in a power-law manner and results in the breakdown of the Fourier heat conduction law in the thermodynamic limit. This has been recognized to be a dominant hydrodynamic effect. Here we show that, instead, the kinetic effect can be dominant in some cases and leads to the Fourier law for finite-size systems. Usually the HCAF undergoes a fast decaying kinetic stage followed by a long slowly decaying hydrodynamic tail. In a finite range of the system size, we find that whether the system follows the Fourier law depends on whether the kinetic stage dominates. Our Rapid Communication is illustrated by the 1D hard-core gas models with which the HCAF is derived analytically and verified numerically by molecular dynamics simulations.

Figure 1

The heat current autocorrelation function of the 1D diatomic gas model at various mass ratios. The solid lines are for simulation results, and the dashed lines of the same color are for the corresponding analytical predictions [Eq. (6)] based on our extended kinetic theory. In simulations, the system size is set to be $N=50000$. Here and in all other figures, $k_B=1,T=1$, and ${\mu }_2=1$. From left to right, the pink, blue, green, and red lines are for $r=1.50,1.30,1.13$, and 1.10, respectively.

Figure 2

The heat conductivity of the 1D diatomic gas model as a function of the system size at various mass ratios. The solid lines are for the results based on the Green-Kubo formula by integrating the heat current autocorrelation function numerically obtained. The horizontal dashed lines of the same color are for the corresponding result ${\kappa }_k$ [Eq. (8)] due to the pure kinetic effect. From bottom to top, the pink, blue, green, and red lines are for $r=1.50,1.30,1.13$, and 1.10, respectively.

Figure 3

The numerical simulation results of the spatiotemporal correlation functions of heat current density fluctuations of the 1D diatomic model with $r=1.3$ at $t=10$ (the blue solid line) and $t=30$ (the red dashed line), respectively. By tracing the sound mode peaks the sound speed is measured numerically.

Figure 4

Comparison of the time scale $t_1$ in the 1D diatomic gas model that separates the kinetic and hydrodynamic processes obtained by Eq. (9) analytically (the red crisscrosses) and by simulations (the blue bullets).

Figure 5

The heat current autocorrelation function for three variant 1D gas models. (a) Periodic diatom model with repeating ${\mu }_1-{\mu }_2-{\mu }_2$ units. (b) Random diatomic gas model. In (a) and (b), ${\mu }_1=1.2$ and ${\mu }_2=1$. (c) Random gas model with the particle masses uniformly distributed between 1 and 1.4. In all the panels, the red dashed lines are for the theoretical prediction given by Eq. (6), and the blue solid lines are for the simulation result.