Large-scale electrical and thermal currents in ordinary metals are well approximated by effective medium theory: global transport properties are governed by the solution to homogenized coupled diffusion equations. In some metals, including the Dirac fluid of nearly charge neutral graphene, microscopic transport is not governed by diffusion, but by a more complicated set of linearized hydrodynamic equations, which form a system of degenerate elliptic equations coupled with the Stokes equation for fluid velocity. In sufficiently inhomogeneous media, these hydrodynamic equations reduce to homogenized diffusion equations. We re-cast the hydrodynamic transport equations as the infimum of a functional over conserved currents, and present a functional framework to model and compute the homogenized diffusion tensor relating electrical and thermal currents to charge and temperature gradients. We generalize to this system two well-known results in homogenization theory: Tartar's proof of local convergence to the homogenized theory in periodic and highly oscillatory media, and sub-additivity of the above functional in random media with highly oscillatory, stationary and ergodic coefficients.

中文翻译：

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狄拉克流体中流体动力传输的均质化

有效介质理论很好地近似了普通金属中的大规模电流和热电流：全局传输特性由均匀耦合扩散方程的解控制。在一些金属中，包括几乎带电的中性石墨烯的狄拉克流体，微观传输不受扩散控制，而是由一组更复杂的线性流体动力学方程控制，这些方程形成了一个退化椭圆方程系统，与流体速度的斯托克斯方程相结合。在足够不均匀的介质中，这些流体动力学方程简化为均匀的扩散方程。我们将流体动力学输运方程重新定义为守恒电流泛函的下界，并提出了一个功能框架来模拟和计算将电流和热电流与电荷和温度梯度相关的均质扩散张量。我们将均匀化理论中的两个众所周知的结果推广到该系统：Tartar 证明在周期性和高度振荡介质中局部收敛到均匀化理论，以及上述函数在具有高度振荡、平稳和遍历系数的随机介质中的次可加性。