Equilibration and Universal Heat Conduction in Fermi-Pasta-Ulam Chains

It is shown numerically that for Fermi-Pasta-Ulam (FPU) chains with alternating masses and heat baths at slightly different temperatures at the ends, the local temperature (LT) on small scales behaves paradoxically in steady state. This expands the long established problem of equilibration of FPU chains. A well-behaved LT appears to be achieved for equal mass chains; the thermal conductivity is shown to diverge with chain length $N$ as $N^{1/3}$, relevant for the much debated question of the universality of one-dimensional heat conduction. The reason why earlier simulations have obtained systematically higher exponents is explained.

Figure 1

Oscillations of the kinetic temperature profile for FPU-$\beta$ dimer chains with mass ratio 2.62, $\gamma =2$. (Top inset) Full profile for a system with $N=128$ and $T_L=2.0$, $T_R=0.5$. (Top) $N=128$ particles, $T_{L,R}=2\pm \Delta T/2$. The scaled temperature is $(T-2)/\Delta T$; the different plots line up. (Bottom) $(T_{2i}-T_{2i+1})N^{1/2}$ versus $2i/N$ for different $N$. $T_{L,R}=2.0$, 0.5. The $N^{1/2}$ and $2i/N$ are chosen to approximately match the vertical and horizontal scales of the plots. Together, the figures imply $\delta T\approx [\Delta T/\sqrt{N}]f(i/N)$.

Figure 2

Kinetic temperature profile for a FPU-$\beta$ chain with $N=16384$, $T_L=2.0$, $T_R=0.5$, and $\gamma =2.0$. The first three even moments of the velocity are shown; their agreement indicates a Gaussian velocity distribution. (Inset) Normalized temperature profiles for different $N$.

Figure 3

Heat current as a function of $N$. Root mean square errors from $O({10}^5-{10}^6)$ measurements are shown except when they are smaller than the points. (Top) Conductivity versus $N$. The last five points fit to a slope of $\alpha =0.333\pm 0.004$. The baths are described in Fig. 2. (Bottom) $j{N}^{1-\alpha }$ versus $N$ for $\alpha =1/3$ and $\alpha =2/5$. In the large $N$ regime, $\alpha$ is definitely less than $2/5$ and appears to agree quite well with the $1/3$ prediction for all data sets. Langevin baths with $\gamma =0.4$, 2, and 10, and one data set with Nose-Hoover baths, are shown.