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Reconstruction of heat relaxation index in phonon transport equation

Peiyi Chen, Irene M. Gamba, Qin Li, Li Wang

TL;DR

The paper addresses recovering the frequency-dependent phonon relaxation time $\tau(\omega)$ from surface temperature data in nanoscale materials where Fourier's law fails. It develops a PDE-constrained inverse framework based on the 1D BGK phonon transport equation, connects the diffusive limit to the classical heat equation via the bulk conductivity $\kappa$, and solves the inverse problem with SGD using adjoint-derived Fréchet derivatives. The main contributions include a rigorous formulation of the inverse problem, a detailed gradient computation via an adjoint system, and numerical validation showing successful reconstruction of $\tau(\omega)$ from synthetic pump-probe data and insight into ballistic-diffusive behavior. This work provides a physics-grounded method to extract microscopic phonon relaxation properties in nano-materials, enabling improved characterization of nanoscale heat transport and guiding experimental design.

Abstract

For nano-materials, heat conductivity is an ill-defined concept. This classical concept assumes the validity of Fourier's law, which states the heat flux is proportional to temperature gradient, with heat conductivity used to denote this ratio. However, this macroscopic constitutive relation breaks down at nano-scales. Instead, heat is propagated using phonon transport equation, an ab initio model derived from the first principle. In this equation, a material's thermal property is coded in a coefficient termed the relaxation time ($τ$). We study an inverse problem in this paper, by using material's temperature response upon heat injection to infer the relaxation time. This inverse problem is formulated in a PDE-constrained optimization, and numerically solved by Stochastic Gradient Descent (SGD) method and its variants. In the execution of SGD, Fréchet derivative is computed and Lipschitz continuity is proved. This approach, in comparison to the earlier studies, honors the nano-structure of of heat conductivity in a nano-material, and we numerically verify the break down of the Fourier's law.

Reconstruction of heat relaxation index in phonon transport equation

TL;DR

The paper addresses recovering the frequency-dependent phonon relaxation time from surface temperature data in nanoscale materials where Fourier's law fails. It develops a PDE-constrained inverse framework based on the 1D BGK phonon transport equation, connects the diffusive limit to the classical heat equation via the bulk conductivity , and solves the inverse problem with SGD using adjoint-derived Fréchet derivatives. The main contributions include a rigorous formulation of the inverse problem, a detailed gradient computation via an adjoint system, and numerical validation showing successful reconstruction of from synthetic pump-probe data and insight into ballistic-diffusive behavior. This work provides a physics-grounded method to extract microscopic phonon relaxation properties in nano-materials, enabling improved characterization of nanoscale heat transport and guiding experimental design.

Abstract

For nano-materials, heat conductivity is an ill-defined concept. This classical concept assumes the validity of Fourier's law, which states the heat flux is proportional to temperature gradient, with heat conductivity used to denote this ratio. However, this macroscopic constitutive relation breaks down at nano-scales. Instead, heat is propagated using phonon transport equation, an ab initio model derived from the first principle. In this equation, a material's thermal property is coded in a coefficient termed the relaxation time (). We study an inverse problem in this paper, by using material's temperature response upon heat injection to infer the relaxation time. This inverse problem is formulated in a PDE-constrained optimization, and numerically solved by Stochastic Gradient Descent (SGD) method and its variants. In the execution of SGD, Fréchet derivative is computed and Lipschitz continuity is proved. This approach, in comparison to the earlier studies, honors the nano-structure of of heat conductivity in a nano-material, and we numerically verify the break down of the Fourier's law.

Paper Structure

This paper contains 15 sections, 5 theorems, 67 equations, 8 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Boltzmann model for heat conductance
  3. Phonon transport equation
  4. Parabolic limit in the diffusion regime
  5. Diffusion equation derivation
  6. Existing Models Relating Heat Conductance to Phonon Frequencies
  7. Inverse problem setup and PDE-constrained optimization
  8. Inverse problem setup
  9. Optimization algorithm
  10. Gradient computation
  11. Numerical results
  12. Numerical study of the forward equation
  13. Numerical study of the inverse problem
  14. Numerical study of the loss function
  15. Physical quantities

Key Result

Proposition 2.1

Here we summarize the properties of $\mathcal{L}$ in LL.

Figures (8)

  • Figure 1: Comparison between quasi-ballistic regime and diffusive regime
  • Figure 1: $\tau^\ast$ and $h^\ast\,.$
  • Figure 2: Forward solution snapshots for $\varepsilon = 1\,.$
  • Figure 3: Temperature measurement at the left boundary: $T(t,x=0)\,.$
  • Figure 4: Forward solution snapshots for $\varepsilon = 0.1\,.$
  • ...and 3 more figures

Theorems & Definitions (7)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem 4.1
  • Theorem 4.2
  • Proposition 4.3
  • Proof 1
  • Remark 4.4