# Hydrodynamic limit for a diffusive system with boundary conditions

## TL;DR

This paper establishes the hydrodynamic limit for an anharmonic chain with boundary conditions, showing convergence to isothermal Navier-Stokes equations and deriving thermodynamic laws from the microscopic model.

## Contribution

It introduces a stochastic model with boundary conditions that converges to macroscopic thermodynamic equations, connecting microscopic dynamics to thermodynamic laws.

## Key findings

- Convergence of microscopic dynamics to isothermal Navier-Stokes equations with boundary conditions
- Derivation of the first and second laws of Thermodynamics from the microscopic model
- Model incorporates boundary conditions and stochastic noise to produce correct thermodynamic behavior

## Abstract

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