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Hydrodynamic limit for a chain with thermal and mechanical boundary forces

Tomasz Komorowski, Stefano Olla, Marielle Simon

Abstract

We prove the hydrodynamic limit for a one dimensional harmonic chain with a random flip of the momentum sign. The system is open and subject to two thermostats at the boundaries and to an external tension at one of the endpoints. 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 system of conservative partial differential equations.

Hydrodynamic limit for a chain with thermal and mechanical boundary forces

Abstract

We prove the hydrodynamic limit for a one dimensional harmonic chain with a random flip of the momentum sign. The system is open and subject to two thermostats at the boundaries and to an external tension at one of the endpoints. 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 system of conservative partial differential equations.

Paper Structure

This paper contains 46 sections, 18 theorems, 311 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Open chain of oscillators
  4. Notations
  5. Hydrodynamic limits: statements of the main results
  6. Empirical distributions of the averages
  7. Convergence of the average stretch and momentum
  8. Convergence of the energy density average
  9. Sketches of proof and Equipartition of energy
  10. Consequences of Assumption \ref{['ass1']}
  11. The boundary terms
  12. Estimates in the bulk
  13. Consequence of Assumption \ref{['ass3']}
  14. Proofs of the hydrodynamic limit theorems
  15. Treatment of boundary terms
  16. ...and 31 more sections

Key Result

theorem \oldthetheorem

Assume that the initial distribution of the stretch and momentum weakly converges to $r_0(\cdot),p_0(\cdot)$ introduced above, i.e. for any test function $G\in C^\infty({\mathbb I})$ we have Then, under Assumption ass1, for any $t>0$ the following holds: weakly in $L^2({\mathbb I})$, where $r(\cdot)$ is the solution of eq:linear--eq:bc0. In addition, we have

Figures (1)

  • Figure 1: An arrow from A to B means that A is used to prove B, but is not necessarily a direct implication.

Theorems & Definitions (32)

  • theorem \oldthetheorem: Convergence of the stretch and momentum profiles
  • theorem \oldthetheorem: Convergence of the mechanical energy profile
  • theorem \oldthetheorem: Convergence of the total energy profile
  • proposition 1
  • proposition 2
  • remark 1
  • proposition 3: $L^2$ bound on average momenta and stretches
  • proposition 4: Energy bound
  • proposition 5: Equipartition of energy
  • lemma 1: Boundary estimates
  • ...and 22 more