Exact diagonal representation of normal mode energy, occupation number, and heat current for phonon-dominated thermal transport

Collective excitations of crystal vibrations or normal modes are customarily described using complex normal mode coordinates. While appropriate for calculating phonon dispersion, the mixed representation involving the complex conjugates does not allow the construction of equivalent phonon occupation number or modal dynamical quantities such as the energy or heat current specific to a wave-vector direction (q). Starting from a canonical solution that includes waves going to the left and right directions, we cast the Hamiltonian, normal mode population, and heat current in an exactly diagonalizable representation using real normal mode amplitudes. We show that the use of real amplitudes obviates the need for a complex modal heat current while making the passage to second quantization more apparent. Using nonequilibrium molecular dynamics simulations, we then compute the net modal energy, heat current, and equivalent phonon population in a linear lattice subjected to a thermal gradient. Our analysis paves a tractable path for probing and computing the direction-dependent thermal-phononic modal properties of dielectric lattices using atomistic simulations.