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Diffusion in Stationary Flow from Mesoscopic Non-equilibrium Thermodynamics

We analyze the diffusion of a Brownian particle in a fluid under stationary flow. By using the scheme of non-equilibrium thermodynamics in phase space, we obtain the Fokker-Planck equation which is compared with others derived from kinetic theory and projector operator techniques. That equation exhibits violation of the fluctuation dissipation-theorem. By implementing the hydrodynamic regime described by the first moments of the non-equilibrium distribution, we find relaxation equations for the diffusion current and pressure tensor, allowing us to arrive at a complete description of the system in the inertial and diffusion regimes. The simplicity and generality of the method we propose, makes it applicable to more complex situations, often encountered in problems of soft condensed matter, in which not only one but more degrees of freedom are coupled to a non-equilibrium bath.