Thermal conductivity for coupled charged harmonic oscillators with noise in a magnetic field

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^1/4for uniformly charged oscillators in dimension 1, while it is known to diverge as t^1/2without 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^1/4and 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.