Heat Conduction and Entropy Production in a One-Dimensional Hard-Particle Gas

We present large scale simulations for a one-dimensional chain of hard-point particles with alternating masses and correct several claims in recent literature based on much smaller simulations. We find heat conductivities $\kappa$ to diverge with the number $N$ of particles. These depended strongly on the mass ratio, and extrapolations to $N\to \infty$, and $t\to \infty$, are difficult due to very large finite-size and finite-time corrections. Nevertheless, our data seem compatible with a universal power law $\kappa \sim N^{\alpha }$ with $\alpha \approx 0.33$ suggesting a relation to the Kardar-Parisi-Zhang model. We finally discuss why the system leads nevertheless to energy dissipation and entropy production, in spite of not being chaotic in the usual sense.

Figure 1

(a) Log-log plot of $J/(T_2-T_1)$ versus $N$ for four values of $r$. Statistical errors are always smaller than the data symbols. (b) Part of the same data divided by $N^{\alpha }$ with $\alpha =0.32$, so the $y$ axis is much expanded.

Figure 2

$T(x)-T_k(x)$ against $x/N$ for $r=1.22$, where $T_k(x)=[T_1^{2/3}-(T_1^{2/3}-T_2^{2/3})x/N{]}^{2/3}$ is the temperature profile according to kinetic theory. In order to reduce statistical fluctuations, we averaged in the curves for $N\ge 1023$ over three successive values of $x$.

Figure 3

$T(x)-T_k(x)$ against $x/N$ for $r=1.618$.

Figure 4

Total heat current autocorrelation, $t^{0.66}{N}^{-1}$ $⟨J(t)J(0)⟩$ for $r=2.2$ and $T=2$. Total momentum is $P=0$.