磁场中含噪声耦合荷电谐振子的热导率
Thermal conductivity for coupled charged harmonic oscillators with noise in a magnetic field
摘要：
我们引入了一个$d$维的带电谐振子系统在磁场中的随机动力学的扰动，其中保存的能量，但不动量。我们通过Green-Kubo公式研究热导率，重点是Green-Kubo积分到时间$t$的渐近行为（即，总能量电流的相关函数的积分）。利用微正则测度计算了均匀带电振子在一般维数$d$下的Green-Kubo公式.我们还发展了一种计算$1$维均匀交替带电振子的Green Kubo公式的方法。证明了导热系数在$1$和$2$维上发散，而在$3$维上保持有限.用微正则系综计算的Green-Kubo积分在$1$维均匀带电振子中发散为$t^{1/4}$，而在无磁场时发散为$t^{1/2}$.这是Green-Kubo积分渐近行为中新指数$1/4$的第一个严格例子。我们还证明，我们的结果提供了第一个严格的例子发散的热导率与消失的声速。此外，利用Green-Kubo公式中的正则测度，证明了均匀带电振子和交替带电振子的Green-Kubo积分分别发散为$t^{1/4}$和$t^{1/2}$ .这意味着指数不仅取决于非零磁场，还取决于振子的电荷结构。
英文摘要：
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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