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

## 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 *t*^{1/4} for uniformly charged oscillators in dimension 1, while it is known to diverge as *t*^{1/2} 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 *t*^{1/4} and *t*^{1/2}. This means that the exponent depends not only on a non-zero magnetic field but also on the charge structure of oscillators.

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## 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.

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