The Fokker-Planck Equation for Lattice Vibration: Stochastic Dynamics and Thermal Conductivity

Abstract

We propose a Fokker-Planck equation (FPE) theory to describe stochastic fluctuation and relaxation processes of lattice vibration at a wide range of conditions, including those beyond the phonon gas (PG) limit. Using the time-dependent, multiple state-variable probability function of a FPE, we first derive time-correlation functions of lattice heat currents in terms of correlation functions among multiple vibrational modes, and subsequently predict the thermal conductivity based on the Green-Kubo formalism. When the kinetic transport theories are valid, this FPE not only predicts a thermal conductivity that is identical to the one predicted by the phonon Boltzmann transport equation, but also provides additional microscopic details on the multiple-mode correlation functions. More importantly, when the kinetic theories become insufficient due to the breakdown of the PG approximation, this FPE theory remains valid to study the correlation functions among vibrational modes in highly anharmonic lattices with significant mode-mode interactions and/or in disordered lattices with strongly localized modes. At the limit of weak mode-mode interactions, we can adopt quantum perturbation theories to derive the drift/diffusion coefficients based on the anharmonicity data derived from first-principles methods. As temperature elevates to the classical regime, we can perform molecular dynamics simulations to directly compute the coefficients. Because these coefficients are defined as ensemble averages at the short time limit, we can implement massive parallel simulation algorithms to take full advantage of the paralleled high-performance computing platforms. A better understanding of the temperature-dependent coefficients up to melting temperatures will provide new insights on microscopic mechanisms that govern the heat conduction through anharmonic and/or disordered lattices beyond the PG model.

References in corpus (12)

• Phonons and related properties of extended systems from density-functional perturbation theory
• Distribution of phonon lifetime in Brillouin zone
• Heat transport in silicon from first principles calculations
• Lattice dynamics of anharmonic solids from first principles
• almaBTE: a solver of the space-time dependent Boltzmann transport equation for phonons in structured materials
• Ab initio variational approach for evaluating lattice thermal conductivity
• Ab initio Green-Kubo Approach for the Thermal Conductivity of Solids
• Microscopic theory and quantum simulation of atomic heat transport
• Thermal transport in crystals as a kinetic theory of relaxons
• On Calculation of Thermal Conductivity from Einstein Relation in Equilibrium MD
• High throughput thermal conductivity of high temperature solid phases: The case of oxide and fluoride perovskites
• Improved Calculation of Vibrational Mode Lifetimes in Anharmonic Solids - Part I: Theory

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