We study the macroscopic profiles of temperature and angular momentum in the stationary state of chains of rotors under a thermo-mechanical forcing applied at the boundaries. These profiles are solutions of a system of diffusive partial differential equations with boundary conditions determined by the thermo-mechanical forcing. Instead of expensive Monte Carlo simulations of the underlying microscopic dynamics, we perform extensive numerical computations based on a finite difference method for the system of partial differential equations describing the macroscopic steady state. We first present a formal derivation of these stationary equations based on a linear response argument and local equilibrium assumptions. We then study various properties of the solutions of these equations. This allows to characterize the regime of parameters leading to uphill energy diffusion—a situation in which the energy flows in the direction of the gradient of temperature—and to identify regions of parameters corresponding to a negative energy conductivity (i.e. a positive linear response of the energy current to a gradient of temperature). The macroscopic equations we derive are consistent with some previous results obtained by numerical simulation of the microscopic physical system, which confirms their validity.

我们研究了在边界处施加热机械力的情况下，转子链的稳态下温度和角动量的宏观分布。这些分布图是具有由热机械强迫确定的边界条件的扩散偏微分方程组的解。代替昂贵的基本微观动力学的蒙特卡洛模拟，我们基于有限差分方法对描述宏观稳态的偏微分方程组进行了广泛的数值计算。我们首先基于线性响应参数和局部均衡假设，对这些平稳方程式进行形式化推导。然后，我们研究这些方程解的各种性质。负能量传导率（即能量电流对温度梯度的正线性响应）。我们得出的宏观方程与通过微观物理系统的数值模拟获得的一些先前的结果一致，这证实了它们的有效性。