Hydrodynamic limit for a diffusive system with boundary conditions

We study the hydrodynamic limit for isothermal dynamics of an anharmonic chain under hyperbolic space-time scaling and with nonvanishing viscosity. The temperature is kept constant by a contact heat bath, realised via stochastic momentum-preserving noise added to dynamics. designed so it contributes macroscopic limit. Dirichlet boundary conditions are also considered: one end fixed, while time-varying tension applied other end. Moreover, Neumann in such way that system produces correct thermodynamic entropy show volume stretch momentum converge (in appropriate sense) smooth solution parabolic conservation laws (isothermal Navier-Stokes equations Lagrangian coordinates) conditions. Finally, changing external allows us define transformations between equilibrium states. use this deduce first second law Thermodynamics our model.