##### Date

2017

##### Dewey

Probabilités et mathématiques appliquées

##### Sujet

hypocoercivity; relative entropy; isothermal transformations; non-equilibrium stationary states; langevin heat bath; Hydrodynamic limits

##### Journal issue

Annals of Probability

##### Volume

45

##### Number

6A

##### Publication date

2017

##### Article pages

3987-4018

##### Publisher

Institute of Mathematical Statistics
##### Author

Letizia, Viviana

Olla, Stefano

##### Type

Article accepté pour publication ou publié

##### Abstract (EN)

We consider a chain of anharmonic oscillators immersed in a heat bath with a temperature gradient and a time-varying tension applied to one end of the chain while the other side is fixed to a point. We prove that under diffusive space-time rescaling the volume strain distribution of the chain evolves following a non-linear diffusive equation. The stationary states of the dynamics are of non-equilibrium and have a positive entropy production, so the classical relative en-tropy methods cannot be used. We develop new estimates based on entropic hypocoercivity, that allow to control the distribution of the position configurations of the chain. The macroscopic limit can be used to model isothermal thermodynamic transformations between non-equilibrium stationary states. 1. Introduction. Macroscopic isothermal thermodynamic transformations can be modeled microscopically by putting a system in contact with Langevin heat bath at a given temperature β −1. In [9] a chain of n anhar-monic oscillators is immersed in a heat bath of Langevin thermostats acting independently on each particle. Macroscopically equivalent isothermal dynamics is obtained by elastic collisions with an external gas of independent particles with Maxwellian random velocities with variance β −1. The effect is to quickly renew the velocities distribution of the particles, so that at any given time it is very close to a maxwellian at given temperature. The chain is pinned only on one side, while at the opposite side a force (tension) τ is acting. The equilibrium distribution is characterized by the two control parameters β −1 , τ (temperature and tension). The total length and the energy of the system in equilibrium are in general non-linear functions of these parameters given by the standard thermodynamic relations. By changing the tension τ applied to the system, a new equilibrium state, with the same temperature β −1 , will be eventually reached. For large n, while the heat bath equilibrates the velocities at the corresponding temperature at time of order 1, the system converges to this global equilibrium length at a time scale of order n 2 t. In [9] it is proven that the length stretch of the system