Unsteady ballistic heat transport: linking lattice dynamics and kinetic theory

The kinetic theory is widely used in the description of thermal transport at the micro- and nanoscale. In the theory, it is assumed that heat is carried by quasi-particles, obeying the Boltzmann transport equation. These quasi-particles are sometimes associated with phonons. However, since phonons are not localized in physical space, they cannot play the role of the quasi-particles used in the kinetic theory. In the present paper, we employ another interpretation of quasi-particles, namely wave packets. Our derivation is carried out for an infinite harmonic chain with a given initial temperature distribution. An exact formula describing the time evolution of the kinetic temperature of the chain is derived. A transition from the exact solution to a continuum limit is performed. It is shown that the resulting continuum solution coincides with the solution of the collisionless Boltzmann equation. The transition yields wave packets, localized in space and moving with group velocities. Therefore, the wave packets can be associated with quasi-particles. It is shown that the quasi-particle has finite lifetime and that its position is determined with some uncertainty.