Advertisement

Thermal Conductivity for Coupled Charged Harmonic Oscillators with Noise in a Magnetic Field

Article
  • 5 Downloads

Abstract

We introduce a d-dimensional system of charged harmonic oscillators in a magnetic field perturbed by a stochastic dynamics which conserves energy but not momentum. We study the thermal conductivity via the Green–Kubo formula, focusing on the asymptotic behavior of the Green–Kubo integral up to time t (i.e., the integral of the correlation function of the total energy current). We employ the microcanonical measure to calculate the Green–Kubo formula in general dimension d for uniformly charged oscillators. We also develop a method to calculate the Green–Kubo formula with the canonical measure for uniformly and alternately charged oscillators in dimension 1. We prove that the thermal conductivity diverges in dimension 1 and 2 while it remains finite in dimension 3. The Green–Kubo integral calculated with the microcanonical ensemble diverges as t1/4 for uniformly charged oscillators in dimension 1, while it is known to diverge as t1/2 without magnetic field. This is the first rigorous example of the new exponent 1/4 in the asymptotic behavior for the Green–Kubo integral. We also demonstrate that our result provides the first rigorous example of a diverging thermal conductivity with vanishing sound speed. In addition, employing the canonical measure in the Green–Kubo formula, we prove that the Green–Kubo integral for uniformly and alternately charged oscillators respectively diverges as t1/4 and t1/2. This means that the exponent depends not only on a non-zero magnetic field but also on the charge structure of oscillators.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The authors express their sincere thanks to Herbert Spohn and Stefano Olla for insightful discussions, and to Shuji Tamaki for careful reading of the manuscript and helpful comments. The authors also thank anonymous referees for their constructive comments that significantly improve the manuscript. KS was supported by JSPS Grants-in-Aid for Scientific Research No. JP26400404 and No. JP16H02211. MS was supported by JSPS Grant-in-Aid for Young Scientists (B) Nos. JP25800068 and JP16KT0021.

References

  1. 1.
    Basile G., Bernardin C., Olla S.: Thermal conductivity for a momentum conservative model. Commun. Math. Phys. 287, 67–98 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Basile G., Bernardin C., Jara M., Komorowski T., Olla S.: Thermal conductivity in harmonic lattices with random collisions. Lecture Notes in Physics 921, 215–237 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bernardin C., Gonçalves P.: Anomalous fluctuations for a perturbed Hamiltonian system with exponential interactions. Commun. Math. Phys. 325, 291–332 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bernardin C., Gonçalves P., Jara M.: 3/4-fractional superdiffusion in a system of harmonic oscillators perturbed by a conservative noise. Arch. Rational Mech. Anal. 220, 505–542 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bernardin C., Stoltz G.: Anomalous diffusion for a class of systems with two conserved quantities. Nonlinearity 25, 1099–1133 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dhar A.: Heat transport in low-dimensional systems. Adv. Phys. 57, 457 (2008)ADSCrossRefGoogle Scholar
  7. 7.
    Jara M., Komorowski T., Olla S.: Superdiffusion of energy in a chain of harmonic oscillators with noise. Commun. Math. Phys. 339, 407–453 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Johnson B.R., Hirschfelder J.O., Yang K.: Interaction of atoms, molecules, and ions with constant electric and magnetic fields. Rev. Mod. Phys. 55, 109 (1983)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Komorowski T., Olla S.: Diffusive propagation of energy in a non-acoustic chain. Arch. Rational Mech. Anal. 223, 95–139 (2017)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Thermal Transport in Low Dimensions: From Statistical Physics to Nanoscale Heat Transfer, Edited by Lepri, S., Springer (2016)Google Scholar
  11. 11.
    Lepri S., Livi R., Politi A.: Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377, 1 (2003)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Narayan O., Ramaswamy S.: Anomalous heat conduction in one-dimensional momentum-conserving systems. Phys. Rev. Lett. 89, 200601 (2002)ADSCrossRefGoogle Scholar
  13. 13.
    Popkov V., Schadschneider A., Schmidt J., Schütz G.M.: Fibonacci family of dynamical universality classes. Proc. Natl. Acad. Sci. 112(41), 12645–12650 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Protter Philip E.: Stochastic Integration and Differential Equations. Springer, Heidelberg (2005)CrossRefMATHGoogle Scholar
  15. 15.
    Tamaki S., Sasada M., Saito K.: Heat transport via low-dimensional systems with broken time-reversal symmetry. Phys. Rev. Lett. 119, 110602 (2017)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Spohn H.: Nonlinear fluctuating hydrodynamics for anharmonic chains. J. Stat. Phys. 154, 1191–127 (2014)Google Scholar
  17. 17.
    Spohn H., Stoltz G.: Nonlinear fluctuating hydrodynamics in one dimension: the case of two conserved fields. J. Stat. Phys. 160, 861–884 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhysicsFaculty of Science and Technology, Keio UniversityYokohamaJapan
  2. 2.Graduate School of Mathematical SciencesUniversity of TokyoTokyoJapan

Personalised recommendations