Bridging the continuum and the kinetic-Boltzmann theories of heat flow through generalized Knudsen numbers

Condensed Matter > Materials Science [Submitted on 16 Jun 2026] Title:Bridging the continuum and the kinetic-Boltzmann theories of heat flow through generalized Knudsen numbers View PDFAbstract:Heat conduction in semiconductor crystals is fundamentally governed by the linearized Peierls-Boltzmann equation (LPBE) for phonon transport, that arises out of a kinetic theory for phonon quasiparticles. Yet, continuum theories such as the Fourier's heat diffusion, weakly quasiballistic and hydrodynamic heat equations are often used to explain the experimental observations of heat flow in these materials. Here, we show that a systematic reduction of the LPBE into such equivalent continuum descriptions are possible only for the limiting values of a set of generalized Knudsen numbers. We further show that all of these continuum heat flow regimes, along with the ballistic heat flow, can be described by a single continuum equation for the temperature field that originates from the eigenmode analysis of the LPBE, thus offering a unified picture of all possible heat flow regimes in semiconducting crystals. Using quantitative examples on twenty three technologically important semiconductors, we show that several previously-unidentified features of the non-Fourier heat flow regimes emerge from this generalized Knudsen number framework such as (1) the mutual exclusivity of the weakly quasiballistic and the hydrodynamic heat flow regimes, (2) length-dependent velocity of the hydrodynamic second sound temperature wave and a characteristic heating length for the strongest hydrodynamic second sound, (3) characteristic frequency-domain temperature response distinguishing the hydrodynamic second sound from the ballistic heat flow regime and, (4) a new non-oscillatory signature of transient hydrodynamic heat flow. Our work formally bridges the continuum and the particulate descriptions of heat flow, and provides insights into the important signatures of temperature dynamics in each of these heat flow regimes, that will aid in their unambiguous experimental observations in the future. Current browse context: cond-mat.mtrl-sci References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) IArxiv Recommender (What is IArxiv?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.