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

Abstract

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 non-interacting particles is here shown to extend to the general case of truly many-body systems. Our approach does not only provide a 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 ± 0.004, in perfect agreement with the dynamical renormalization-group prediction (1/3).PACS numbers: 44.10.+i,05.45.Jn, 05.70.Ln After the discovery of anomalous heat conductivity in classical one-dimensional lattice systems [1], in the last years a renewed attention has been devoted to the old problem of identifying the minimal ingredients required for the Fourier law to be ensured. As summarized in a recent review article [2], many different models have been numerically investigated to identify the physical conditions under which the thermal conductivity κ diverges with the system size L and, having assessed that κ ≈ L α , to determine the possibly different universality classes for the divergence rate α. Simultaneously, several attempts have been made to estimate analytically the scaling behaviour of κ: self-consistent mode-coupling theory [2] and the Boltzmann equation [3] suggest that α = 2/5, while dynamical renormalization group indicates α = 1/3 [4]. Both predictions are compatible with numerical simulations which are, however, often affected by relatively strong finite-size corrections. The only system where convincing results have been obtained is the Fermi, Pasta Ulam β-model in the infinite temperature limit. Its behaviour is consistent with α = 2/5 [5], but the simmetry of the potential casts doubts about the generality of this model [6]. A further simple system that can be effectively simulated on a computer is the diatomic hard-point gas: there, interactions are provided by elastic collisions of point-like particles [7]. Unfortunately, the most detailed numerical simulations reported in the literature show a slow growth of the divergence rate with L, so that some authors claim that α = 1/3 [8], while others state that the conservative guess α = 0.25 is more realistic [9]. Settling this issue is not only conceptually important, but it is a necessary requisite to later quantify finite-size corrections, a crucial issue in applications to, e.g., carbon nanotubes, where one needs to know the prefactor as well.Although the problem involves intrinsically many degrees of freedom, some researchers have tried to shed some light with reference to the simpler setup of noninteracting particles moving along a periodic array of convex scatterers (billiard gas channels) [10]. The absence of interactions simplifies the task of understanding heat conductivity and allows, in particular, tracing b...

Energy diffusion in hard-point systems

“…Actually, this latter interpretation would make the model almost identical to that one studied in [33]. The quantitative result of numerical studies are very similar to the diatomic HPG [16] and to the hard-point chain model with ℓ = 0.2. More precisely, the evolution in the tangent space is in a good agreement with the Levy-walk model with the same scaling exponent γ = 0.6 (Fig.…”

Heat transport in low-dimensional systems

“…However, based on the theoretical predictions, there seems reason to believe that the value obtained by Grassberger et al is the correct one and here we will discuss their results in some detail. We also mention here the work of Cipriani et al [110] who performed zero-temperature studies on diffusion of localized pulses and using a Levy walk interpretation concluded that α = 0.333 ± 0.004. We now present some of the results obtained by Grassberger et al [108].…”

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

“…(1) is satisfied with α = 1/3. This has been confirmed by simulations of hard particle gases [16,17,18], although very large systems are required [19] and the issue is not completely settled [20]. On the other hand, numerical simulations of oscillator chains, including FPU chains, give various exponents [6,10,11,12,13] for different systems, often slightly higher than 1/3.…”