From Anomalous Energy Diffusion to Levy Walks and Heat Conductivity in One-Dimensional Systems

The evolution of infinitesimal, localized perturbations is investigated in a one-dimensional diatomic gas of hard-point particles (HPG) and thereby connected to energy diffusion. As a result, a Levy walk description, which was so far invoked to explain anomalous heat conductivity in the context of noninteracting particles is here shown to extend to the general case of truly many-body systems. Our approach does not only provide firm evidence that energy diffusion is anomalous in the HPG, but proves definitely superior to direct methods for estimating the divergence rate of heat conductivity which turns out to be $0.333\pm 0.004$, in perfect agreement with the dynamical renormalization-group prediction ($1/3$).

Figure 1

Spectrum of comoving Lyapunov exponents at different times for $r=3$; the chain length is $N=8190$. From bottom to top the curves have been obtained by comparing perturbations and times $(100,200)$, $(200,400)$, $(400,800)$, $(800,1200)$, and $(800,1600)$. The perturbation amplitude has been averaged over ${10}^4$ different realizations.

Figure 2

Maximum height ${\delta }_{(2)}(0,t)$ of the infinitesimal perturbations profile (circles) and mean square displacement ${\sigma }^2(t)$ (diamonds) versus time.

Figure 3

Rescaled perturbation profiles at $t=40$, 80, 160, 320, 640, 1280, 2560, and 3840 (the width increases with time), for $\gamma =3/5$. The profiles have been obtained by averaging over ${10}^4$ realizations. In the inset, the profile at $t=640$ (solid line) is compared with the propagators of a Levy walk for an exponent $\mu =\frac{5}{3}$ with a fixed velocity $u=1$ (dotted line) and a Gaussian distribution with average equal to 1 and rms 0.036 (dashed line). In the last two cases, the propagators have been obtained by averaging over ${10}^8$ different realizations.