Perturbative treatment of inter-site couplings in the local description of open quantum networks

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A system of sites weakly coupled to each other and to one or more reservoirs (open quantum network) is considered. A new quantum master equation which improves shortcomings of the master equation known in the so-called local approach is derived. The usual quantum master equation describing the weak coupling of the system with reservoir requires the knowledge of eigenvalues and eigenvectors of the Hamiltonian of the system, so it often becomes impractical. By this reason, when the inter-site couplings are weak, the local approach, which neglects the influence of the inter-site couplings on the system-reservoir couplings, is often used. However, recently, it was reported that the local approach master equation leads to the violation of the second law of thermodynamics. We develop a systematic perturbation expansion to derive corrections to the local approach master equation. Using this improvement of the local approach, we derive an expression for the heat flux for a particular model and show that it does not violate the second law of thermodynamics.