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We consider a chain of particles connected by an-harmonic springs, with aboundary force (tension) acting on the last particle, while the first particleis kept pinned at a point. The particles are in contact with stochastic heatbaths, whose action on the dynamics conserves the volume and the momentum,while energy is exchanged with the heat baths in such way that, in equilibrium,the system is at a given temperature $T$. We study the space empirical profilesof volume stretch and momentum under hyperbolic rescaling of space and time asthe size of the system growth to be infinite, with the boundary tensionchanging slowly in the macroscopic time scale. We prove that the probabilitydistributions of these profiles concentrate on $L^2$-valued weak solutions ofthe isothermal Euler equations (i.e. the non-linear wave equation, also calledp-system), satisfying the boundary conditions imposed by the microscopicdynamics. Furthermore, the weak solutions obtained satisfy the Clausiusinequality between the work done by the boundary force and the change of thetotal free energy in the system. This result includes the shock regime of thesystem.

Hydrodynamic Limits and Clausius inequality for Isothermal Non-linear Elastodynamics with Boundary Tension