Towards a Microscopic Derivation of the Phonon Boltzmann Equation

The thermal conductivity of insulating (dielectric) crystals is computed almost exclusively on the basis of the phonon Boltzmann equation. We refer to [1] for a discussion more complete than possible in this contribution. On the microscopic level the starting point is the Born-Oppenheimer approximation (see [2] for a modern version), which provides an effective Hamiltonian for the slow motion of the nuclei. Since their deviation from the equilibrium position is small, one is led to a wave equation with a weak nonlinearity. As already emphasized by R. Peierls in his seminal work [3], physically it is of importance to retain the structure resulting from the atomic lattice, which forces the discrete wave equation. On the other hand, continuum wave equations with weak nonlinearity appear in the description of the waves in the upper ocean and in many other fields. This topic is referred to as weak turbulence. Again the theoretical treatment of such equations is based mostly on the phonon Boltzmann eqation, see e.g. [4]. In these applications one considers scales which are much larger than the atomistic scale, hence quantum effects are negligible. For dielectric crystals, on the other side, quantum effects are of importance at low temperatures. We refer to [1] and discuss here only the classical discrete wave equation with a small nonlinearity. If one considers crystals with a single nucleus per unit cell, then the displacement field is a 3-vector field over the crystal lattice Γ. The nonlinearity results from the weakly non-quadratic interaction potentials between the nuclei. As we will see, the microscopic mechanism responsible for the validity of the Boltzmann equation can be understood already in case the displacement field is declared to be scalar, the nonlinearity to be due to an on-site potential, and the lattice Γ = Z. This is the model I will discuss in my notes. As the title indicates there is no complete proof available for the validity of the phonon Boltzmann equation. The plan is to explain the kinetic scaling and to restate our conjecture in terms of the asymptotics of certain Feynman diagrams.