We establish a connection between anomalous heat conduction and anomalous diffusion in one-dimensional systems. It is shown that if the mean square of the displacement of the particle is $⟨\Delta x^2⟩=2D{t}^{\alpha }(0<\alpha \le 2)$, then the thermal conductivity can be expressed in terms of the system size $L$ as $\kappa =c{L}^{\beta }$ with $\beta =2-2/\alpha$. This result predicts that normal diffusion ($\alpha =1$) implies normal heat conduction obeying the Fourier law ($\beta =0$) and that superdiffusion ($\alpha >1$) implies anomalous heat conduction with a divergent thermal conductivity ($\beta >0$). More interestingly, subdiffusion ($\alpha <1$) implies anomalous heat conduction with a convergent thermal conductivity ($\beta <0$), and, consequently, the system is a thermal insulator in the thermodynamic limit. Existing numerical data support our results.

We study anomalous heat conduction and anomalous diffusion in low-dimensional systems ranging from nonlinear lattices, single walled carbon nanotubes, to billiard gas channels. We find that in all discussed systems, the anomalous heat conductivity can be connected with the anomalous diffusion, namely, if energy diffusion is σ2(t)=2Dtα(0<α⩽2), then the thermal conductivity can be expressed in terms of the system size L as κ=cLβ with β=2−2∕α. This result predicts that a normal diffusion (α=1) implies a normal heat conduction obeying the Fourier law (β=0), a superdiffusion (α>1) implies an anomalous heat conduction with a divergent thermal conductivity (β>0), and more interestingly, a subdiffusion (α<1) implies an anomalous heat conduction with a convergent thermal conductivity (β<0), consequently, the system is a thermal insulator in the thermodynamic limit. Existing numerical data support our theoretical prediction.

We study heat conduction behavior of one-dimensional lattices with asymmetric, momentum conserving interparticle interactions. We find that with a certain degree of interaction asymmetry, the heat conductivity measured in nonequilibrium stationary states converges in the thermodynamical limit. Our analysis suggests that the mass gradient resulting from asymmetric interactions may provide a phonon scattering mechanism in addition to that caused by nonlinear interactions.

We study in momentum-conserving systems, how nonintegrable dynamics may affect thermal transport properties. As illustrating examples, two one-dimensional (1D) diatomic chains, representing 1D fluids and lattices, respectively, are numerically investigated. In both models, the two species of atoms are assigned two different masses and are arranged alternatively. The systems are nonintegrable unless the mass ratio is one. We find that when the mass ratio is slightly different from one, the heat conductivity may keep significantly unchanged over a certain range of the system size and as the mass ratio tends to one, this range may expand rapidly. These results establish a new connection between the macroscopic thermal transport properties and the underlying dynamics.

We study diffusion processes of local fluctuations of heat, energy, momentum, and mass in three paradigmatic one-dimensional systems. For each system, diffusion processes of four physical quantities are simulated and the cross correlations between them are investigated. We find that, in all three systems, diffusion processes of energy and mass can be perfectly expressed as a linear combination of those of heat and momentum, suggesting that diffusion processes of heat and momentum may represent the heat mode and the sound mode in the hydrodynamic theory. In addition, the dynamic structure factor, which describes the diffusion behavior of local mass density fluctuations, is in general insufficient for probing diffusion processes of other quantities because in some cases there is no correlation between them. We also find that the diffusion behavior of heat can be qualitatively different from that of energy, and, as a result, previous studies trying to relate heat conduction to energy diffusion should be revisited.

We study energy transport in a one-dimensional model of elastically colliding particles with alternate masses $m$ and $M$. In order to prevent total momentum conservation, we confine particles with mass $M$ inside a cell of finite size. We provide convincing numerical evidence for the validity of Fourier law of heat conduction in spite of the lack of exponential dynamical instability. Comparison with previous results on similar models shows the relevance of the role played by total momentum conservation.