Non-Fourier heat conduction : from phase-lag models to relativistic and quantum transport

著者

    • Zhmakin, Alexander I.

書誌事項

Non-Fourier heat conduction : from phase-lag models to relativistic and quantum transport

Alexander I. Zhmakin

Springer, c2023

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注記

Includes bibliographical references

内容説明・目次

内容説明

This book presents a broad and well-structured overview of various non-Fourier heat conduction models. The classical Fourier heat conduction model is valid for most macroscopic problems. However, it fails when the wave nature of the heat propagation becomes dominant and memory or non-local spatial effects become significant; e.g., during ultrafast heating, heat transfer at the nanoscale, in granular and porous materials, at extremely high values of the heat flux, or in heat transfer in biological tissues. The book looks at numerous non-Fourier heat conduction models that incorporate time non-locality for materials with memory, such as hereditary materials, including fractional hereditary materials, and/or spatial non-locality, i.e. materials with a non-homogeneous inner structure. Beginning with an introduction to classical transport theory, including phase-lag, phonon, and thermomass models, the book then looks at various aspects of relativistic and quantum transport, including approaches based on the Landauer formalism as well as the Green-Kubo theory of linear response. Featuring an appendix that provides an introduction to methods in fractional calculus, this book is a valuable resource for any researcher interested in theoretical and numerical aspects of complex, non-trivial heat conduction problems.

目次

1. Introduction Part I Classical Transport 2. Phase-Lag Models 2.1 Maxwell-Cattaneo-Vernotte Equation 2.2. Dual-Phase-Lag Model 2.2.1. Nonlocal Dual-Phase-Lag Model 2.3.Triple-Phase-Lag Model 2.3.1. Nonlocal Triple-Phase-Lag Model 3. Phonon Models 3.1. Guyer-Krumhansl (GK) Equation 3.2. Ballistic-Diffusive Model 3.3. Unified Nondiffusive-Diffusive Model 3.4. Two-fluid model 3.5. Generalized Fourier Law by Hua et al. 3.6. Phonon hydrodynamics 3.7. Relaxon Model 4. Thermomass Model 4.1. Equation of State (EOS) of the Thermon Gas 4.1.1. EOS of thermon gas in ideal gas 4.1.2. EOS of thermon gas in dielectrics 4.1.3. EOS of thermon gas in metals 4.2. Equations of Motion of Thermon Gas 4.3. Heat flow Choking Phenomenon 4.4. Dispersion of Thermal Waves 5. Mesoscopic Moment Equations Thermodynamic Models 6.1. Jou & Cimmelli Model 6.2. Kov\'acs & V\'an Model 6.3. Rogolino et al. Models 6.4. EIT Ballistic-Diffusive Model 7. Fractional Models 7.1. Fractional Fourier Model 7.1.1. Nonlinear Diffusivity 7.1.2. Fractional Pennes model 7.2. Zingales's Fractional Order Model 7.3. Fractional Cattaneo and SPL Models 7.4. Fractional DPL Model 7.5. Fractional TPL Model 7.5.1. Nonlocal Fractional TPL Model 8. Fractional Boltzmann and Fokker-Planck equations 8.1. Continuous Time Random Walks 8.1.1. L\'evy (Khintchine-L\'evy) walks 8.2. Kramers-Fokker-Plank equation 8.3. Li & Cao Model 9. Elasticity and thermal expansion coupling 9.1. Non-Fourier Thermoelasticity 9.1.1. Fractional Thermoelasticity 10. Some Exact Solutions 10.1. Phase-Lag Models 10.2. Phonon Models 10.3. Fractional Models Part II Relativistic Transport *** 1 *** 2 Part III Quantum Transport 13. Landauer approach 14. Green-Kubo approach 15. Phonon Coherent Transport 16. Conclusions 17. Appendix. An Introduction to Fractional Calculus 17.1. Fractional Derivatives 17.1.1. Riemann-Liouville Fractional Integral 17.1.2. Riemann-Liouville Fractional Derivative 17.1.2.1. Leibniz' formula 17.1.2.2. Fa\'a di Bruno formula (chain rule) 17.1.2.3. Fractional Taylor expansion 17.1.2.4. Symmetrised space derivative 1 7.1.3. Caputo Fractional Derivative 17.1.4. Caputo & Fabrizio Fractional Derivatives 17.1.5. GC & GRL derivatives 17.1.5.1. GC derivatives 17.1.5.2. GRL derivatives 17.1.6. Marchaud-Hadamard Fractional Derivatives 17.1.7. Gr\"unwald - Letnikov Derivative 17.1.8. Riesz Fractional Operators 17.1.9. Weyl Fractional Derivative 17.1.10. Erd\'elye-Kober Fractional Operators 17.1.11. Interpretation of Fractional Integral and Derivatives 17.1.12. Local Fractional Derivatives 17.1.12.1. "Conformable" Fractional Derivative 17.2. Fractional Differential Equations 17.2.1. Distributed order differential equations 17.2.2 Special Functions 17.2.2.1. Mittag-Leffler Functions 17.2.2.2. H Functions 17.2.2.3. Wright Functions 17.3. Solution of Fractional Differential Equations 17.3.1. Analytic Methods 17.3.2. Numerical Methods

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