Fokker-Planck equation for lattice vibration: Stochastic dynamics and thermal conductivity

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 limit. Using the time-dependent, multiple state-variable probability function of a vibration FPE, we first derive time-correlation functions of lattice heat currents in terms of correlation functions among multiple vibrational modes, and subsequently predict the lattice thermal conductivity based on the Green-Kubo formalism. When the quasiparticle kinetic transport theories are valid, this vibration FPE not only predicts a lattice 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 phonon gas 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 lattice anharmonicity data derived from first-principles methods. As temperature elevates to the classical regime, we can perform molecular dynamics simulations to directly compute the drift/diffusion coefficients. Because these coefficients are defined as ensemble averages at the limit of $\delta t\to 0$, 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 drift/diffusion 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 phonon gas model.

Figure 1

Schematic plot of lattice thermal conductivity ${\kappa }_{\text{Latt}}$ as a function of temperature $T$, up to the melting temperature $T_{\text{melt}}$.