Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Thermal boundary conditions in fractional superdiffusion of energy

Tomasz Komorowski, Stefano Olla

TL;DR

The paper rigorously derives the hydrodynamic limit for energy transport in a finite, unpinned one-dimensional chain with stochastic momentum exchange and Langevin baths, showing that the macroscopic energy density evolves via a nonlocal Lévy-type operator with explicit boundary kernels. It identifies the boundary layer structure and provides a precise mathematical formulation of absorption, reflection, and transmission of long-wavelength phonons at the baths, distinguishing the boundary behavior from diffusive and pinned settings. The authors develop a comprehensive entropy-covariance framework, resolve the covariance matrix, and perform delicate bulk-boundary asymptotics to prove existence and uniqueness of the limiting weak solution, including boundary conditions enforced by the baths. The results offer the first rigorous derivation of fractional-boundary conditions arising from local microscopic dynamics and illuminate how microscopic bilinear interactions yield macroscopic nonlocal boundary behavior with potential relevance to experimental and numerical studies of fractional energy diffusion in low-dimensional systems.

Abstract

We study heat conduction in a one-dimensional {finite}, unpinned chain of atoms perturbed by stochastic momentum exchange and coupled to Langevin heat baths at {possibly} distinct temperatures placed at the endpoints of the chain. While infinite systems without boundaries are known to exhibit superdiffusive energy transport described by a fractional heat equation with the generator $-|Δ|^{3/4}$, the corresponding boundary conditions induced by heat baths remain less understood. We establish the hydrodynamic limit for a finite chain with $n+1$ atoms connected to thermostats at the endpoints, deriving the macroscopic evolution of the averaged energy profile. The limiting equation is governed by a non-local Lévy-type operator, with boundary terms determined by explicit interaction kernels that encode absorption, reflection, and transmission of long-wavelength phonons at the baths. Our results provide the first rigorous identification of boundary conditions for fractional superdiffusion arising directly from microscopic dynamics with local interactions, highlighting their distinction from both diffusive and pinned-chain settings

Thermal boundary conditions in fractional superdiffusion of energy

TL;DR

The paper rigorously derives the hydrodynamic limit for energy transport in a finite, unpinned one-dimensional chain with stochastic momentum exchange and Langevin baths, showing that the macroscopic energy density evolves via a nonlocal Lévy-type operator with explicit boundary kernels. It identifies the boundary layer structure and provides a precise mathematical formulation of absorption, reflection, and transmission of long-wavelength phonons at the baths, distinguishing the boundary behavior from diffusive and pinned settings. The authors develop a comprehensive entropy-covariance framework, resolve the covariance matrix, and perform delicate bulk-boundary asymptotics to prove existence and uniqueness of the limiting weak solution, including boundary conditions enforced by the baths. The results offer the first rigorous derivation of fractional-boundary conditions arising from local microscopic dynamics and illuminate how microscopic bilinear interactions yield macroscopic nonlocal boundary behavior with potential relevance to experimental and numerical studies of fractional energy diffusion in low-dimensional systems.

Abstract

We study heat conduction in a one-dimensional {finite}, unpinned chain of atoms perturbed by stochastic momentum exchange and coupled to Langevin heat baths at {possibly} distinct temperatures placed at the endpoints of the chain. While infinite systems without boundaries are known to exhibit superdiffusive energy transport described by a fractional heat equation with the generator , the corresponding boundary conditions induced by heat baths remain less understood. We establish the hydrodynamic limit for a finite chain with atoms connected to thermostats at the endpoints, deriving the macroscopic evolution of the averaged energy profile. The limiting equation is governed by a non-local Lévy-type operator, with boundary terms determined by explicit interaction kernels that encode absorption, reflection, and transmission of long-wavelength phonons at the baths. Our results provide the first rigorous identification of boundary conditions for fractional superdiffusion arising directly from microscopic dynamics with local interactions, highlighting their distinction from both diffusive and pinned-chain settings
Paper Structure (89 sections, 51 theorems, 673 equations)

This paper contains 89 sections, 51 theorems, 673 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dynamics in the stretch/momentum configuration space
  4. Fractional diffusion equation with Dirichlet boundary conditions
  5. Scaled dynamics of the chain
  6. The macroscopic limit of the energy functional
  7. Energy currents
  8. Some bounds on entropy and covariances
  9. Proof of Theorem \ref{['entropy-t']} in the case $T_L=T_R$
  10. Estimates of some covariances
  11. Covariance matrix
  12. Preliminaries
  13. Resolution of the covariance matrix
  14. Further covariance bounds from \ref{['energy-bal']}
  15. Limit identification. Formulation of the results
  16. ...and 74 more sections

Key Result

Theorem 2.3

Suppose that $T_{\rm ini}\in L^2[0,1]$. Then, equation f-diff1 has a unique solution $T(\cdot,\cdot)$. In addition, the solution satisfies

Theorems & Definitions (55)

  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Theorem 2.9
  • Corollary 2.10
  • Theorem 2.13: The limit of thermal energy and equipartition
  • Theorem 2.14
  • ...and 45 more