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Register a new userAnomalous heat conduction and anomalous diffusion in nonlinear lattices, single walled nanotubes, and billiard gas channels
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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 $sigma^2(t)equiv <Delta x^2> =2Dt^{alpha} (0<alphale 2)$, then the thermal conductivity can be expressed in terms of the system size $L$ as $kappa = cL^{beta}$ with $beta=2-2/alpha$. This result predicts that a normal diffusion ($alpha =1$) implies a normal heat conduction obeying the Fourier law ($beta=0$), a superdiffusion ($alpha>1$) implies an anomalous heat conduction with a divergent thermal conductivity ($beta>0$), and more interestingly, a subdiffusion ($alpha <1$) implies an anomalous heat conduction with a convergent thermal conductivity ($beta<0$), consequently, the system is a thermal insulator in the thermodynamic limit. Existing numerical data support our theoretical prediction.
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We study heat conduction in one dimensional (1D) anharmonic lattices analytically and numerically by using an effective phonon theory. It is found that every effective phonon mode oscillates quasi-periodically. By weighting the power spectrum of the total heat flux in the Debye formula, we obtain a unified formalism that can explain anomalous heat conduction in momentum conserved lattices without on-site potential and normal heat conduction in lattices with on-site potential. Our results agree very well with numerical ones for existing models such as the Fermi-Pasta-Ulam model, the Frenkel-Kontorova model and the $phi^4$ model etc.
Heat transport in one-dimensional (1D) momentum-conserving lattices is generally assumed to be anomalous, thus yielding a power-law divergence of thermal conductivity with system length. However, whether heat transport in two-dimensional (2D) system is anomalous or not is still on debate because of the difficulties involved in experimental measurements or due to the insufficiently large simulation size. Here, we simulate energy and momentum diffusion in the 2D nonlinear lattices using the method of fluctuation correlation functions. Our simulations confirm that energy diffusion in the 2D momentum-conserving lattices is anomalous and can be well described by the L{e}vy-stable distribution. We also find that the disappear of side peaks of heat mode may suggest a weak coupling between heat mode and sound mode in the 2D nonlinear system. It is also observed that the harmonic interactions in the 2D nonlinear lattices can accelerate the energy diffusion. Contrary to the hypothesis of 1D system, we clarify that anomalous heat transport in the 2D momentum-conserving system cannot be corroborated by the momentum superdiffusion any more. Moreover, as is expected, lattices with a nonlinear on-site potential exhibit normal energy diffusion, independent of the dimension. Our findings offer some valuable insights into the mechanism of thermal transport in 2D system.
This study is concerned with destruction of Anderson localization by a nonlinearity of the power-law type. We suggest using a nonlinear Schrodinger model with random potential on a lattice that quadratic nonlinearity plays a dynamically very distinguished role in that it is the only type of power nonlinearity permitting an abrupt localization-delocalization transition with unlimited spreading already at the delocalization border. For super-quadratic nonlinearity the borderline spreading corresponds to diffusion processes on finite clusters. We have proposed an analytical method to predict and explain such transport processes. Our method uses a topological approximation of the nonlinear Anderson model and, if the exponent of the power nonlinearity is either integer or half-integer, will yield the wanted value of the transport exponent via a triangulation procedure in an Euclidean mapping space. A kinetic picture of the transport arising from these investigations uses a fractional extension of the diffusion equation to fractional derivatives over the time, signifying non-Markovian dynamics with algebraically decaying time correlations.
We introduce a model for diffusion of two classes of particles ($A$ and $B$) with priority: where both species are present in the same site the motion of $A$s takes precedence over that of $B$s. This describes realistic situations in wireless and communication networks. In regular lattices the diffusion of the two species is normal but the $B$ particles are significantly slower, due to the presence of the $A$ particles. From the fraction of sites where the $B$ particles can move freely, which we compute analytically, we derive the diffusion coefficients of the two species. In heterogeneous networks the fraction of sites where $B$ is free decreases exponentially with the degree of the sites. This, coupled with accumulation of particles in high-degree nodes leads to trapping of the low priority particles in scale-free networks.
Recent investigations call attention to the dynamics of anomalous diffusion and its connection with basic principles of statistical mechanics. We present here a short review of those ideas and their implications.
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