Abstract
We prove the hydrodynamic limit for an harmonic chain with a random exchange of momentum that conserves the kinetic energy but not the momentum. The system is open and subject to two thermostats at the boundaries and to external tension. Under a diffusive scaling of space-time, we prove that the empirical profiles of the two locally conserved quantities, the volume stretch and the energy, converge to the solution of a non-linear diffusive systems of conservative partial differential equations.
Highlights
The mathematical derivation of the macroscopic evolution of the conserved quantities of a physical system, from its microscopic dynamics, through a rescaling of space and time has been the subject of much research in the last 40 years
Heuristic assumptions like local equilibrium and linear response permit to formally derive the macroscopic equations [14], mathematical proofs are very difficult and most of the techniques used are based on relative entropy methods
In some situations a different approach, based on Wigner distributions, is effective in controlling the macroscopic evolution of energy. This is the case for a chain of harmonic springs with a random flip of sign of the velocities, provided with periodic boundary conditions, for which the total energy and the total length of the system are the two conserved quantities, and where the hydrodynamic limit has been proven in [10]
Summary
The mathematical derivation of the macroscopic evolution of the conserved quantities of a physical system, from its microscopic dynamics, through a rescaling of space and time (so called hydrodynamic limit) has been the subject of much research in the last 40 years (cf. [9] and references within). Concerning the hydrodynamic limit, in the appendix section of [11] we have formulated a heuristic argument, based on entropy production estimates, that has not been proved there, and that substantiated the validity of the macroscopic equations governing the dynamics in the case of a random momentum exchange microscopic model, see Section 2.2 of [11]. In order to control these terms we need to bound the rate of damping of the mechanical energy, which is done in Lemma 5.4 These controls allow us to prove that the L2 norm of the covariances of random fluctuations of momenta and stretches, at the given time, grows with the logarithm of the size of the system: this is the content of Proposition 8.1. In the appendix sections we give the proofs of quite technical estimates used throughout Section 6
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