We consider a one-dimensional chain of coupled oscillators in contact at both ends with heat baths at different temperatures, and subject to an external force at one end. The Hamiltonian dynamics in the bulk is perturbed by random exchanges of the neighbouring momenta such that the energy is locally conserved. We prove that in the stationary state the energy and the volume stretch profiles, in large scale limit, converge to the solutions of a diffusive system with Dirichlet boundary conditions. As a consequence the macroscopic temperature stationary profile presents a maximum inside the chain higher than the thermostats temperatures, as well as the possibility of uphill diffusion (energy current against the temperature gradient). Finally, we are also able to derive the non-stationary macroscopic coupled diffusive equations followed by the energy and volume stretch profiles.

中文翻译：

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热传导的开放微观模型：演化和非平衡稳态

我们考虑一维耦合振荡器链，其两端与不同温度的热浴接触，并在一端受到外力。体中的哈密顿动力学受到相邻动量的随机交换的干扰，因此能量是局部守恒的。我们证明，在静止状态下，能量和体积拉伸剖面在大尺度限制下收敛到具有狄利克雷边界条件的扩散系统的解。因此，宏观温度稳定分布在链内呈现出高于恒温器温度的最大值，以及向上扩散的可能性（能量流与温度梯度相反）。最后，