Abstract
We consider a d-dimensional disordered harmonic chain (DHC) perturbed by an energy conservative noise. We obtain uniform in the volume upper and lower bounds for the thermal conductivity defined through the Green-Kubo formula. These bounds indicate a positive finite conductivity. We prove also that the infinite volume homogenized Green-Kubo formula converges.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aoki, K., Lukkarinen, J., Spohn, H.: Energy transport in weakly anharmonic chains. J. Stat. Phys. 124(5), 1105–1129 (2006)
Basile, G., Bernardin, C., Olla, S.: Momentum conserving model with anomalous thermal conductivity in low dimensional systems. Phys. Rev. Lett. 96, 204303 (2006)
Basile, G., Bernardin, C., Olla, S.: Thermal conductivity for a momentum conserving model. Commun. Math. Phys. (2008, to appear). http://arxiv.org/abs/cond-mat/0601544
Basile, G., Delfini, L., Lepri, S., Livi, R., Olla, S., Politi, A.: Anomalous transport and relaxation in classical one-dimensional models. Eur. Phys. J. Spec. Top. 151, 85 (2007)
Benabou, G.: Homogenization of Ornstein-Uhlenbeck process in random environment. Commun. Math. Phys. 266, 699–714 (2006)
Bernardin, C.: Hydrodynamics for a system of harmonic oscillators perturbed by a conservative noise. Stoch. Process. Appl. 117, 487–513 (2007)
Bernardin, C., Olla, S.: Fourier’s law for a microscopic heat conduction model. J. Stat. Phys. 121, 271–289 (2005)
Bonetto, F., Lebowitz, J.L., Lukkarinen, J.: Fourier’s law for a harmonic crystal with self-consistent stochastic reservoirs. J. Stat. Phys. 116, 783–813 (2004)
Bonetto, F., Lebowitz, J.L., Lukkarinen, J., Olla, S.: Private communication
Bricmont, J., Kupinianen, A.: Towards a derivation of Fourier’s law for coupled anharmonic oscillators. Commun. Math. Phys. 274(3), 555–626 (2007)
Casher, A., Lebowitz, J.L.: Heat flow in regular and disordered harmonic chains. J. Math. Phys. 12, 1701 (1971)
Ethier, S.N., Kurtz, T.G.: Markov Processes. Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York (1986)
Funaki, T., Spohn, H.: Motion by mean curvature from the Ginzburg-Landau ∇ φ interface model. Commun. Math. Phys. 185, 1–36 (1997)
Lepri, S., Livi, R., Politi, A.: Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377, 1–80 (2003)
Landim, C., Yau, H.T.: Fluctuation-dissipation equation of asymmetric simple exclusion process. Probab. Theory Relat. Fields 108(3), 321–356 (1997)
Dhar, A.: Heat conduction in the disordered harmonic chain revisited. Phys. Rev. Lett. 86, 5882 (2001)
Dhar, A., Lebowitz, J.L.: Effect of phonon-phonon interactions on localization. arXiv:0708.4171 (2007)
Lukkarinen, J., Spohn, H.: Anomalous energy transport in the FPU-beta chain. arXiv:0704.1607 (2007)
Rieder, Z., Lebowitz, J.L., Lieb, E.: Properties of harmonic crystal in a stationary non-equilibrium state. J. Math. Phys. 8, 1073–1078 (1967)
Roy, D., Dhar, A.: Role of pinning potentials in heat transport through disordered harmonic chain (2008). http://arxiv.org/abs/0806.4693v1
Rubin, R.J., Greer, W.L.: Abnormal lattice thermal conductivity of a one-dimensional. harmonic, isotopically disordered crystal, J. Math. Phys. 12(8), 1686–1701 (1971)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bernardin, C. Thermal Conductivity for a Noisy Disordered Harmonic Chain. J Stat Phys 133, 417–433 (2008). https://doi.org/10.1007/s10955-008-9620-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-008-9620-1