%0 Book Section %1 statphys23_0579 %A Shiba, H. %A Yukawa, S. %A Ito, N. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K conduction dynamics formula green-kubo heat lattice long-time molecular nonlinear simulation statphys23 tails topic-3 %T Anomalous Heat Conduction in Three-dimensional Nonlinear Lattices %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=579 %X Recently, heat conduction in one-dimensional nonlinear oscillator chains has been investigated. Many strenuous studies have elucidated that thermal conductivity of such a system with momentum conservation depends on the size of the system as $\kappaN^\alpha$ where the exponent $\alpha$ takes values near 0.4. This property seems to hold in many nonlinear lattice systems with nonintegrability such as FPU chains, diatomic Toda lattices, etc. The results mean that thermal conductivity of such systems does not have any finite value in the thermodynamic limit $N\rightarrowınfty$. The anomalous behavior is thought to originate in a slow decay of the equilibrium autocorrelation function of heat flux, called ''long-time tails''. This decay is typically represented by a power-law function as \ J(t)J(0)t^-d/2.\qquad \ Here, $J(t)$ is the total heat flux and $d$ is the dimensionality of the system. This type of decay is known to widely hold in dense gases and liquids, and momentum conservation is thought to play an important role for such a slow decay. In this paper, we investigate heat conduction in simple three-dimensional extensions of three-dimensional nonlinear lattices. When $d=3$, the usual long-time tails will predict some finite conductivity through Green-Kubo formula. Our starting point is a simple three-dimensional model of which the Hamiltonian is given by \ H = \sum_i=1^N \vecp_i^22m +\sum_i,j\rangle łeft k2|r_i-r_j |^2 +g4 |r_i-r_j|^4 \right . \ We will present that the three-dimensional lattices have diverging thermal conductivity, contrary to a naive expectation. The conductivity seems to show power-law divergence as $\kappaN_z^\alpha$, and the long-time tail of the heat flux autocorrelation function is of a completely different type from the conventional ones. This anomalous behavior is so robust that the conductivity diverges with its size even when the system contains disorders in the mass, etc. Our results evidence that such divergence of thermal conductivity is a robust and universal nature of three-dimensional model insulators. We think that our results is the first case that anomalous conductivity is essentially found in three-dimensional cases, and that the result arises an important problem about what is the origin of finite conductivity which we usually observe in three-dimensional insulators. 1. S. Lepri, R. Livi, and A. Politi: Phys. Rep. 377, 1 (2003), and references therein. 2. H. Shiba, S. Yukawa, and N. Ito: J. Phys. Soc. Jpn. 75, 103001 (2006). We have reported logarithmic divergence of thermal conductivityin this paper. However, our current notion is that the divergence is not of a logarithmic but a weak power-law type. 3. H. Shiba and N. Ito: in preparation.

@incollection{statphys23_0579, abstract = {Recently, heat conduction in one-dimensional nonlinear oscillator chains has been investigated. Many strenuous studies have elucidated that thermal conductivity of such a system with momentum conservation depends on the size of the system as $\kappa\sim N^\alpha$ where the exponent $\alpha$ takes values near 0.4. This property seems to hold in many nonlinear lattice systems with nonintegrability such as FPU chains, diatomic Toda lattices, etc. The results mean that thermal conductivity of such systems does not have any finite value in the thermodynamic limit $N\rightarrow\infty$. The anomalous behavior is thought to originate in a slow decay of the equilibrium autocorrelation function of heat flux, called ''long-time tails''. This decay is typically represented by a power-law function as \[ \langle J(t)\cdot J(0)\rangle \sim t^{-d/2}.\qquad \] Here, $J(t)$ is the total heat flux and $d$ is the dimensionality of the system. This type of decay is known to widely hold in dense gases and liquids, and momentum conservation is thought to play an important role for such a slow decay. In this paper, we investigate heat conduction in simple three-dimensional extensions of three-dimensional nonlinear lattices. When $d=3$, the usual long-time tails will predict some finite conductivity through Green-Kubo formula. Our starting point is a simple three-dimensional model of which the Hamiltonian is given by \[ H = \sum_{i=1}^N \frac{\vec{p}_i^2}{2m} +\sum_{\langle i,j\rangle} \left[ \frac{k}{2}|\vec{r}_i-\vec{r}_j |^2 +\frac{g}{4} |\vec{r}_i-\vec{r}_j|^4 \right] . \] We will present that the three-dimensional lattices have diverging thermal conductivity, contrary to a naive expectation. The conductivity seems to show power-law divergence as $\kappa\sim N_z^\alpha$, and the long-time tail of the heat flux autocorrelation function is of a completely different type from the conventional ones. This anomalous behavior is so robust that the conductivity diverges with its size even when the system contains disorders in the mass, etc. Our results evidence that such divergence of thermal conductivity is a robust and universal nature of three-dimensional model insulators. We think that our results is the first case that anomalous conductivity is essentially found in three-dimensional cases, and that the result arises an important problem about what is the origin of finite conductivity which we usually observe in three-dimensional insulators. 1. S. Lepri, R. Livi, and A. Politi: Phys. Rep. 377, 1 (2003), and references therein. 2. H. Shiba, S. Yukawa, and N. Ito: J. Phys. Soc. Jpn. 75, 103001 (2006). We have reported logarithmic divergence of thermal conductivityin this paper. However, our current notion is that the divergence is not of a logarithmic but a weak power-law type. 3. H. Shiba and N. Ito: in preparation.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Shiba, H. and Yukawa, S. and Ito, N.}, biburl = {https://www.bibsonomy.org/bibtex/2c0ef6959df8a65b1905bca077bbd3004/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {7016cbe20927139a1e5470323bee8a62}, intrahash = {c0ef6959df8a65b1905bca077bbd3004}, keywords = {conduction dynamics formula green-kubo heat lattice long-time molecular nonlinear simulation statphys23 tails topic-3}, month = {9-13 July}, timestamp = {2007-06-20T10:16:24.000+0200}, title = {Anomalous Heat Conduction in Three-dimensional Nonlinear Lattices}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=579}, year = 2007 }