Heat transport and current fluctuations in harmonic crystals

• This article is a pedagogic review on heat transport across a harmonic crystal.
• Heat baths are modeled through generalized quantum Langevin equations.
• The Landauer formula for phonon heat transport is derived.
• Applications of the Landauer formula to ordered and disordered harmonic chains is discussed.
• Current fluctuations are studied, again using the Langevin equations approach, and a formula for the cumulant generating function is derived.

The approach of studying heat conduction in harmonic crystals, using generalized Langevin equations and phonon Green’s functions, is explained in this review. We present the detailed derivation of the Landauer formula for phonon heat transfer using this approach. Some applications of the Landauer formula to understanding heat conduction in one-dimensional ordered and mass-disordered harmonic chains are given. It is explained how different boundary conditions lead to completely different system-size dependence of the average current in disordered chains. Finally we discuss the fluctuations in the total heat transferred in a fixed large interval of time and show how the Langevin approach is also useful in computing properties of these fluctuations in a harmonic system. In particular we derive the general formula for the cumulant generating function of these fluctuations.