I will present a way to derive, via hydrodynamic limits, weak solutions to the 1D isothermal Euler
equations in Lagrangian coordinates. This is obtained from a microscopic anharmonic chain with momentum
preserving noise and hyperbolic scaling. Boundary conditions are added so we can define thermodynamic
transformations between macroscopic equilibrium states: in particular we study the first and the second
law of thermodynamics for the macroscopic system.