Hydrodynamic limits under pressure

We study the hydrodynamic limit for a one dimensional isothermal anharmonic finite chain in Lagrangian coordinates with hyperbolic space-time scaling. The temperature is kept constant by putting the chain in contact with a heath bath, realised via the addition of a stochastic momentum-preserving noise to the dynamics of the chain. The noise is designed to be large at the microscopic level, but vanishing in the hydrodynamic limit. Boundary conditions are also considered: one end of the chain is kept fixed, while a time-variable tension is applied to the other end. We show that the microscopic deformation and momentum converge (in an appropriate sense) to solutions of a system of hyperbolic conservation laws (isothermal Euler equations in Lagrangian coordinates) with boundary conditions. Since these solutions may develop shocks in a finite time, they are obtained in a weak sense. This is done by adapting the theory of compensated compactness to a stochastic setting. Finally, the external tension allows us to define thermodynamic transformations between equilibrium states. We use this to deduce the first and the second principle of Thermodynamics for our model.