Abstract We derive a large-scale hydrodynamic equation, including diffusive and dissipative effects, for systems with generic static position-dependent driving forces coupling to local conserved quantities. show that this equation predicts entropy increase thermal states as the only stationary states. The applies any system number of local, parity time-symmetric quantities, in arbitrary dimensi...

Wave propagation and diffusion in linear materials preserve local reciprocity terms of a symmetric Green's function. For wave propagations, the relation between fields entering leaving system is more relevant than detailed information about inside it. In such cases, global scattering off through several ports important, which defined as transmission channels. When two-port supports nonreciproca...