🔬 mesoscale physics

greenWTE: Frequency-domain solver for the phonon Wigner Transport Equation with arbitrary heating using Green's functions

The paper introduces **greenWTE**, a Python package that solves the phonon Wigner Transport Equation in the frequency domain using Green's functions to model heat transport with arbitrary heating, enabling the calculation of thermal conductivities and system responses across scales from bulk to nanoscale by accounting for both phonon scattering and tunneling processes.

Original authors: Laurenz Kremeyer, Bradley J. Siwick, Samuel Huberman

Published 2026-11-04

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CC BY 4.0

Original authors: Laurenz Kremeyer, Bradley J. Siwick, Samuel Huberman

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Heat moves through solid materials in ways that are usually easy to predict. When you touch a hot pan, the warmth travels to your hand following a simple, well-known rule that scientists have used for over a century. This rule works perfectly for large objects and slow changes, treating heat like a smooth, flowing river. However, when scientists look at heat moving through tiny materials or over incredibly short moments, that simple rule breaks down. At these microscopic scales, heat does not flow smoothly; instead, it behaves more like a collection of individual particles that can bounce, scatter, and even pass through each other in strange ways. Understanding this behavior is crucial for designing better electronics and new materials, but it requires a much more complex way of looking at the problem.

For decades, researchers have used a standard method to describe how these heat-carrying particles move, a method that works well for materials that conduct heat efficiently. Yet, in materials that are poor conductors, this standard approach misses a vital piece of the puzzle. It fails to account for a subtle quantum effect where particles can act like waves, allowing them to tunnel through barriers or interfere with one another. To truly understand heat in these difficult materials, scientists needed a new mathematical framework that could capture both the particle-like collisions and these wave-like behaviors. Until now, calculating how heat moves under these complex conditions, especially when the heating changes rapidly or comes from specific spots, has been a significant challenge.

A team of researchers at McGill University has developed a new computer tool called greenWTE to solve this problem. This software allows scientists to simulate how heat moves through materials by solving a complex equation that tracks both the position and the speed of heat-carrying particles, while also accounting for their wave-like nature. The tool is designed to handle any kind of heating source, whether it is a steady stream of warmth or a sudden, intense burst of energy. By using a mathematical technique that breaks the problem down into waves of different frequencies, the software can predict how a material will respond to heat in both space and time. This approach bridges the gap between theoretical physics and real-world experiments, such as those that use ultrafast lasers to heat materials for a fraction of a second and then measure the result.

The researchers built this tool as a flexible program that can run on standard computers or, for faster results, on powerful graphics processors. It offers two main ways to find the answer: one method calculates a complete map of how the material responds to any possible heat source, which is powerful but requires a lot of computing power to set up; the other method solves the problem directly for a specific heating scenario without building the full map first. This flexibility means scientists can now model experiments where heat is applied in specific patterns, like a ring or a spot, or where energy is injected into specific types of vibrations within the material. The software can also take the results from these frequency-based calculations and convert them back into a timeline, allowing for a direct comparison with experiments that measure heat as it happens over time.

The impact of this work lies in its ability to reveal how heat behaves in materials that were previously too difficult to study. By simulating the transition from simple, smooth heat flow to complex, wave-dominated movement, the tool helps researchers understand why some materials conduct heat so poorly. This knowledge is essential for improving devices that rely on managing heat at the nanoscale. The authors have made the software available to the scientific community, providing a way to test ideas about microscopic heat transport that were previously only theoretical. With this tool, scientists can now design better experiments and interpret their results with a deeper understanding of the invisible, rapid dance of energy within solids.

Technical Summary of greenWTE: Frequency-domain solver for the phonon Wigner Transport Equation with arbitrary heating using Green's functions

Problem Statement
Heat transport in solids is traditionally modeled phenomenologically via Fourier's law. However, at small time and length scales, particularly in low-thermal-conductivity materials, a microscopic description is required. While the phonon Boltzmann Transport Equation (BTE) has been successful for high-thermal-conductivity materials, it fails to capture phonon tunneling processes (coherences) essential for low-thermal-conductivity systems. Although the Wigner Transport Equation (WTE) was developed to address these coherences, existing software solutions are limited. Open-source tools like Phono3py, ShengBTE, and AlmaBTE are restricted to calculating static bulk thermal conductivities. Conversely, closed-source implementations exist for solving the BTE with arbitrary source terms in Fourier space, but a corresponding open-source solver for the WTE framework that handles arbitrary heating beyond the static limit has been absent.

Methodology
The authors present greenWTE, a Python package designed to solve the Wigner Transport Equation in spatial and temporal Fourier space. The core equation solved is:
$-i\omega \tilde{N}(k, q, \omega) + \frac{i}{\hbar} [\Omega(q), \tilde{N}(k, q, \omega)] + ik \frac{1}{2} \{V(q), \tilde{N}(k, q, \omega)\} = F\left\{\frac{\partial}{\partial t}N(R, q, t)\bigg|_{H_{col}}\right\} + \tilde{Q}(k, q, \omega)$
where $\tilde{N}$ is the Wigner distribution, $\Omega$ is the phonon frequency matrix, $V$ is the velocity operator, and $\tilde{Q}$ represents the source term in Fourier space.

The system is reformulated as a linear operator equation $L(k, \omega)\tilde{N}(k, \omega) = \tilde{Q}(k, \omega)$ . The package offers two distinct numerical approaches to solve this:

Direct Solver: Fully inverts the operator $L$ at each $(k, \omega)$ point to determine the Green's function $G(k, \omega)$ . While computationally expensive initially, this allows for the efficient reuse of the Green's function to compute temperature responses to various user-chosen source terms.
Iterative Solver: Solves the linear system $L\tilde{N} = \tilde{Q}$ directly for a specific source term using an iterative method. This avoids the high cost of full Green's function inversion but requires solving the system from scratch for each new source term.

The software is built to integrate with Phono3py for generating material-specific inputs. It utilizes NumPy for CPU execution and CuPy for GPU acceleration, with the latter highly recommended for handling the computationally intensive linear algebra operations required for realistic scenarios.

Key Contributions and Results

Arbitrary Source Terms: Unlike previous static-limit solvers, greenWTE allows users to define arbitrary source terms $\tilde{Q}$ . This enables the modeling of specific experimental heating conditions, such as Gaussian or ring-like spatial profiles, and the selective injection of energy into specific Brillouin zone regions or phonon modes.
Bridge to Experiments: By solving in the frequency domain, the solver naturally connects to ultrafast pump-probe experiments (where heating is quasi-instantaneous) and transient thermal grating experiments (involving spatially periodic temperature gratings).
Output Capabilities: The package outputs the Wigner distribution $\tilde{N}(k, q, \omega)$ and derived integral quantities, including heat flux and thermal conductivity. These calculations span from bulk to nanoscale regimes and cover both static and dynamic thermal conductivities.
Implementation: The code is open-source, written in Python, and includes documentation via ReadTheDocs. It supports the calculation of responses to physically realistic source terms derived from electron-phonon and phonon-phonon coupling calculations associated with ultrafast laser radiation.

Significance and Research Impact
The paper positions greenWTE as a tool to fill the gap between microscopic theory and experimental observation in thermal transport. Its primary significance lies in enabling the study of the transition from population-dominated to coherence-dominated nondiffusive thermal transport in low-thermal-conductivity materials. By allowing the simulation of pump-probe experiments through tailored source terms, the software aims to facilitate better interpretation of experimental results and a deeper understanding of microscopic thermal transport phenomena at short time and length scales. The authors note that time-domain response functions can be obtained by Fourier transforming the frequency-domain results, allowing for direct comparison with time-domain experiments. The work is presented as a step toward generalizing these approaches to arbitrary experimental and device geometries.

Technical Summary of greenWTE: Frequency-domain solver for the phonon Wigner Transport Equation with arbitrary heating using Green's functions

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