### 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 amethod to calculate the Green–Kubo formulawith 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.

Original language | English |
---|---|

Pages (from-to) | 951-995 |

Number of pages | 45 |

Journal | Communications in Mathematical Physics |

Volume | 361 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2018 Jan 1 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**Thermal conductivity for coupled charged harmonic oscillators with noise in a magnetic field.** / Saitou, Keiji; Sasada, Makiko.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 361, no. 3, pp. 951-995. https://doi.org/10.1007/s00220-018-3198-5

}

TY - JOUR

T1 - Thermal conductivity for coupled charged harmonic oscillators with noise in a magnetic field

AU - Saitou, Keiji

AU - Sasada, Makiko

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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 amethod to calculate the Green–Kubo formulawith 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.

AB - 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 amethod to calculate the Green–Kubo formulawith 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.

UR - http://www.scopus.com/inward/record.url?scp=85056110989&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85056110989&partnerID=8YFLogxK

U2 - 10.1007/s00220-018-3198-5

DO - 10.1007/s00220-018-3198-5

M3 - Article

AN - SCOPUS:85056110989

VL - 361

SP - 951

EP - 995

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -