随机能量交换系统中傅立叶定律的推导
On the derivation of Fourier's law in stochastic energy exchange systems
摘要：
我们提出了一个详细的推导傅立叶定律在一类随机能量交换系统，自然地表征二维力学系统的局部约束粒子的相互作用。随机系统由一系列能量变量组成，这些能量变量可以在最近的邻居之间以可变的速率进行部分交换。我们提供了两个独立的热导率的推导，并证明这个量是相同的能量交换的频率。第一个推导依赖于Helfand矩的扩散，其仅由静态平均值确定。第二种方法依赖于一个非平衡稳态周围的概率测度的梯度展开。热流的线性部分由求解玻尔兹曼方程的局部热平衡分布确定。数值计算的电导率沿我们的两种方法。结果与我们的理论非常吻合。
英文摘要：
We present a detailed derivation of Fourier's law in a class of stochastic energy exchange systems that naturally characterize two-dimensional mechanical systems of locally confined particles in interaction. The stochastic systems consist of an array of energy variables which can be partially exchanged among nearest neighbours at variable rates. We provide two independent derivations of the thermal conductivity and prove this quantity is identical to the frequency of energy exchanges. The first derivation relies on the diffusion of the Helfand moment, which is determined solely by static averages. The second approach relies on a gradient expansion of the probability measure around a non-equilibrium stationary state. The linear part of the heat current is determined by local thermal equilibrium distributions which solve a Boltzmann-like equation. A numerical scheme is presented with computations of the conductivity along our two methods. The results are in excellent agreement with our theory.
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