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Heat flow in a periodically forced, unpinned thermostatted chain

Tomasz Komorowski, Stefano Olla, Marielle Simon

TL;DR

The paper establishes a diffusive hydrodynamic limit for a one-dimensional unpinned harmonic chain with velocity-flip noise, coupled to a Langevin heat bath on the left and driven by a time-periodic force on the right. It derives macroscopic equations for the stretch $r(t,u)$ and energy $e(t,u)$, including a nonlinear diffusion for the energy and a Neumann-type boundary condition at the driven edge that incorporates a boundary work term $W^Q$ from fast forcing. The authors prove convergence of the macroscopic stretch and mechanical energy, decompose the total work into a mechanical part and a thermal contribution, and provide a full energy-covariance analysis showing equipartition and local equilibrium in the diffusive limit. The results quantify how mechanical energy is transformed into thermal energy and how boundary forcing contributes to macroscopic energy transport, under entropy bounds and a decorrelation assumption for lower thermal modes. This advances the understanding of energy transport and energy-thermodynamic distinctions in noisy, boundary-driven harmonic chains.

Abstract

We prove the hydrodynamic limit for a one-dimensional harmonic chain of interacting atoms with a random flip of the momentum sign. The system is open: at the left boundary it is attached to a heat bath at temperature $T_-$, while at the right endpoint it is subject to an action of a force which reads as $\bar F + \frac 1{\sqrt n} \widetilde{\mathcal F} (n^2 t)$, where $\bar F \ge0$ and $\widetilde{\mathcal F}(t)$ is a periodic function. Here $n$ is the size of the microscopic system. 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, as $n\to+\infty$, to the solution of a non-linear diffusive system of conservative partial differential equations with a Dirichlet type and Neumann boundary conditions on the left and the right endpoints, respectively.

Heat flow in a periodically forced, unpinned thermostatted chain

TL;DR

The paper establishes a diffusive hydrodynamic limit for a one-dimensional unpinned harmonic chain with velocity-flip noise, coupled to a Langevin heat bath on the left and driven by a time-periodic force on the right. It derives macroscopic equations for the stretch and energy , including a nonlinear diffusion for the energy and a Neumann-type boundary condition at the driven edge that incorporates a boundary work term from fast forcing. The authors prove convergence of the macroscopic stretch and mechanical energy, decompose the total work into a mechanical part and a thermal contribution, and provide a full energy-covariance analysis showing equipartition and local equilibrium in the diffusive limit. The results quantify how mechanical energy is transformed into thermal energy and how boundary forcing contributes to macroscopic energy transport, under entropy bounds and a decorrelation assumption for lower thermal modes. This advances the understanding of energy transport and energy-thermodynamic distinctions in noisy, boundary-driven harmonic chains.

Abstract

We prove the hydrodynamic limit for a one-dimensional harmonic chain of interacting atoms with a random flip of the momentum sign. The system is open: at the left boundary it is attached to a heat bath at temperature , while at the right endpoint it is subject to an action of a force which reads as , where and is a periodic function. Here is the size of the microscopic system. 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, as , to the solution of a non-linear diffusive system of conservative partial differential equations with a Dirichlet type and Neumann boundary conditions on the left and the right endpoints, respectively.
Paper Structure (34 sections, 27 theorems, 344 equations)

This paper contains 34 sections, 27 theorems, 344 equations.

Table of Contents

  1. Introduction
  2. Definition of the dynamics and results
  3. Description of the model
  4. Formulation of results
  5. Macroscopic evolution of stretch and mechanical energy
  6. Macroscopic work
  7. The macroscopic limit of the energy functional
  8. Evolution of the averages in diffusive scaling
  9. Homogeneous evolution and key lemma
  10. Control of the $\ell^2$-norm of momenta averages
  11. Control of the $\ell^2$-norm of the stretches averages
  12. Estimates at the boundaries
  13. Stretch and mechanical energy: Proof of Theorem \ref{['th-r']}
  14. Macroscopic evolution of the stretch
  15. Macroscopic evolution of the mechanical energy
  16. ...and 19 more sections

Key Result

Theorem 2.4

Assume Assumptions ass1 and ass2. Then, for any continuous test function $\varphi:[0,1]\to{\mathbb R}$, $t>0$ and where $r(t,u)$ is the solution of eq:HLstretch.

Theorems & Definitions (41)

  • Definition 2.3
  • Theorem 2.4: Limit of stretch and mechanical energy
  • Theorem 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.10
  • Remark 2.11
  • Definition 2.12
  • Theorem 2.13: Limit of total energy
  • Remark 2.14
  • ...and 31 more