Pressure-induced recovery of Fourier's law in one-dimensional momentum-conserving systems

We report the two typical models of normal heat conduction in one-dimensional momentum-conserving systems. They show the Arrhenius and the non-Arrhenius temperature dependence. We construct the two corresponding phenomenologies, transition-state theory of thermally activated dissociation and the pressure-induced crossover between two fixed points in fluctuating hydrodynamics. Compressibility yields the ballistic fixed point, whose scaling is observed in Fermi-Pasta-Ulam (FPU) $\beta$ lattices.

Figure 1

The temperature profile of $c$-FPU $\beta$ at $(T_L,T_R,l,N)=(50,10,4,2^{15})$. One can see the gaps and the varying gradient.

Figure 2

Bulk heat conductivity (left: PR-6; right: $c$-FPU $\beta$). Parameters are $(T_L,T_R,d)=(4,1,3/4)$ for PR-6 and $(T_L,T_R,\gamma ,l)=(50,30,2,4)$ for $c$-FPU $\beta$. They show the convergence except $c$-FPU $\beta$ at $l=0$. The transient exponents are $1/2$ in PR-6 and $0.33$ in $c$-FPU $\beta$ at $l=4$. For accuracy, we check the power of the last two points in $c$-FPU $\beta$ at $l=2,4$ as an indicator of the convergence. The powers are $0.018(l=2)$ and $0.009(l=4)$, which show the quantitative evidence of the convergence.

Figure 3

Local heat conductivity (left: PR-6; center and right: $c$-FPU $\beta$). Parameters are $(T_L,T_R,d)=(4,1,3/4)$ for PR-6 and $(T_L,T_R,l)=(50,10,4)$ for $c$-FPU $\beta$ in the center panel. The right panel shows the values of FPU $\beta$ at the largest size of our observations with multiplication of normalization factors. They can be fitted with a unique curve approximately. One can observe the two different types of temperature dependence: the Arrhenius type connected with the thermally activated recovery in PR-6 and the non-Arrhenius type in $c$-FPU-$\beta$, suggesting another process from thermal activation.

Figure 4

Left: Normalized power spectra of the total energy currents $|\widehat{J}{\left(\omega \right)|}^2/N$ of $c$-FPU $\beta$ at $T=30$ in ${10}^{-5}\lesssim \omega \lesssim 1$. Right: Comparison between the local heat conductivity at the averaged temperature $\overline{T}$ of steady heat conduction and the corresponding value given by the Green Kubo formula. One can see the anomalous scaling $|\widehat{J}{\left(\omega \right)|}^2/N\sim {\omega }^{-\alpha }$ at small $\omega$ of $l=0$, but Lorentzian $|\widehat{J}{\left(\omega \right)|}^2/N\sim a/[1+{\left(b\omega \right)}^2]$ recovers with $l$ increasing $\left(l=2,3\right)$. There is no system size dependence at $l=2,3$. The trends of $(a,b)$ that $(a,b)$ increase with $l$ are consistent with Fig. 2. Pressure looks to suppress the long time tail. The comparison of heat conductivity (right) shows the consistent convergence of heat conductivity in highly compressed FPU $\beta$.

Figure 5

The RG flow field (43) and (44) of approximated full fluctuating hydrodynamic equations in the passive scalar limit. There are four fixed points: $(\overline{\lambda },\overline{Y})=(0,0),(0,\infty ),(\sqrt{\pi },0),(\sqrt{2\pi },1-2^{-2/3})$. The latter two points are the well-known inviscid fixed point and the nontrivial ballistic fixed point. The linearly stable fixed point is the ballistic fixed point only. The inviscid fixed point is unstable except in $\overline{Y}=0$. We marked the three fixed points taking finite values and show the typical flow in the same figure. $\overline{Y}$ slowly increases around the inviscid fixed point and the flow is accelerated after the detachment and wraps around the ballistic fixed point.

Figure 6

Left: The equilibrium autocorrelation of the heat mode in $c$-FPU $\beta$ at $T=1,t=t_n\sim 400n/c,N=2048$. Right: $S(0,t)$ in $t\le t_2$. $z$ values of the left panel are chosen as $z=3/2$ at $l\ne 2$ and $z=1$ at $l=2$. One can see the symmetric Levy in sound cones at $l=0$ and the growing bumps around the cones at $l\ne 0$. $S(x,t)$ shows the ballistic scaling at $l=2$. The width scaling is the inviscid one $x/t^{2/3}$ at $l=0$ and the ballistic one $x/t$ at $l=2$. $S(0,t)$ shows the clear crossover.

Figure 7

Temperature profile at $c$-FPU $\beta$ at $l=0,\gamma =2,T_L=50,T_R=30[T_{L\left(R\right)}$: left (right) reservoir temperature]. The corresponding curve of normal heat conduction $\kappa \propto T^{1/4}$ is also shown in the same figure. There are the cusps at the edges corresponding to the anomaly but the gaps at the edges cannot be seen. The clear systematic deviation means the asymptotics of the anomaly realized in this parameter region.

Figure 8

Temperature profile at $c$-FPU $\beta$ at $l=4,\gamma =2,T_L=50,T_R=30$. The corresponding curve of normal heat conduction $\kappa \propto T^{-2.3}$ is also shown in the same figure. The clear convergence means the realization of the asymptotic form in this system. The gaps at the edges vanish at larger sizes.